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📝 Differentiability of multivariable functions (14 MCQs)

📖 From Calculus • 14. Partial Derivatives Calculus • 14 questions available

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What is Differentiability of multivariable functions?

Definition:
$f$ is differentiable at $(a,b)$ if $\lim_{(h,k)\to(0,0)} \frac{f(a+h,b+k)-f(a,b)-f_x h - f_y k}{\sqrt{h^2+k^2}} = 0$ , stronger than partial existence.

Example:
$f(x,y) = |x| + |y|$ has partials at origin but fails differentiability due to corner; $f=x^2+y^2$ is differentiable everywhere.

Reason:
This precise definition captures true local linearity, distinguishing smooth surfaces from those with cusps or ridges despite having directional derivatives.

Easy

Medium

Hard

📝 All Differentiability of multivariable functions MCQs

Q1. A function $f(x,y)$ has partial derivatives $f_x(0,0) = 0$ and $f_y(0,0) = 0$ . However, the limit defining differentiability at the origin does not exist. Which statement best explains this phenomenon?

A.The partial derivatives are discontinuous at the origin, which is a necessary condition for non-differentiability.

B.The function fails to be continuous at the origin, making differentiability impossible regardless of partials.

C.Differentiability requires the linear approximation error to vanish faster than the distance, which partials alone cannot guarantee. ✅

D.The directional derivatives in all directions must equal zero for differentiability, but they do not.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Students often confuse existence of partial derivatives with differentiability. Differentiability demands that the tangent plane approximates the surface with an error term that is little-o of the norm. Partial derivatives only capture behavior along axes; pathological behavior along other curves can violate the stricter linear approximation requirement even when axis-aligned slopes exist.

Q2. Consider $f(x,y) = \frac{x^3}{x^2+y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0)=0$ . A student claims it is differentiable at the origin because all directional derivatives exist and equal zero. Identify the flaw in this reasoning.

A.Directional derivatives existing in all directions is sufficient for differentiability if they are all zero.

B.The student failed to verify continuity before checking directional derivatives.

C.Existence of all directional derivatives does not imply the linear map defined by them approximates the function uniformly. ✅

D.The function is actually not continuous at the origin, invalidating the claim.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: This question targets error analysis. While all directional derivatives may exist and equal zero, differentiability requires the remainder term in the Taylor expansion to be $o(\|(x,y)\|)$ . Directional derivatives only test convergence along straight lines; non-linear paths like parabolas may reveal discrepancies that prevent the existence of a true tangent plane.

Q3. Given the contour plot of a smooth surface where level curves near point P are concentric ellipses becoming increasingly dense, what can be inferred about differentiability and gradient magnitude at P?

A.The function is not differentiable at P because the contours are closed loops.

B.The function is differentiable at P and the gradient vector is zero due to symmetry.

C.The function is differentiable at P with a large gradient magnitude perpendicular to the ellipses. ✅

D.Differentiability cannot be determined from contour plots alone without explicit formulas.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Contour density indicates rate of change. Dense, smooth, nested ellipses suggest a well-defined tangent plane and thus differentiability. The gradient points perpendicular to level curves toward increasing values, and its magnitude correlates with contour spacing. Closed smooth contours around a point typically indicate a local extremum or saddle with non-zero gradient unless the center itself is a critical point.

Q4. Let $f(x,y) = |x| + y^2$ . At which points is $f$ differentiable, and why does standard partial derivative testing fail to fully characterize this set?

A.Differentiable everywhere except on the y-axis; partial

f_x

does not exist there, correctly identifying non-differentiability. ✅

B.Differentiable everywhere; absolute value is smoothed by the

y^2

term in multivariable calculus.

C.Differentiable only where

x > 0

; partial derivatives exist on the y-axis but the function lacks a tangent plane there.

D.Differentiable everywhere except at the origin; partials fail to detect issues away from the origin.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: This tests conceptual understanding of how univariate non-smoothness extends to multivariable functions. The term $|x|$ creates a ridge along the entire y-axis where $f_x$ is undefined. Since differentiability implies existence of all partials, non-existence of $f_x$ on the y-axis definitively proves non-differentiability there. Elsewhere, the function is a sum of differentiable components.

Q5. A model for heat distribution uses $T(x,y,t)$ . Engineers approximate $\Delta T \approx T_x \Delta x + T_y \Delta y$ near a sensor. Under what physical condition would this linear approximation produce unacceptable errors despite small $\Delta x, \Delta y$ ?

A.When the temperature field is stationary and partial derivatives are constant.

B.When the sensor is located at a point where

T

is differentiable but second-order terms dominate due to high curvature. ✅

C.When the measurement time interval

\Delta t

is too large relative to spatial changes.

D.When the coordinate system is rotated relative to the principal axes of heat flow.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This application scenario emphasizes that differentiability guarantees good linear approximation only asymptotically. In practice, if second derivatives (curvature) are very large, the quadratic remainder term may be significant even for small displacements. Engineers must assess not just differentiability but also the magnitude of higher-order terms to determine if linearization is practically valid for their tolerance requirements.

Q6. Suppose $f(x,y)$ satisfies $f(h,k) - f(0,0) - ah - bk = \sqrt{|hk|}$ for constants $a,b$ . Is $f$ differentiable at $(0,0)$ ?

A.Yes, because

\sqrt{|hk|} \to 0

(h,k) \to (0,0)

, satisfying the limit definition.

B.No, because

\sqrt{|hk|}

is not

o(\sqrt{h^2+k^2})

; the ratio does not tend to zero. ✅

C.Yes, because the expression defines a valid linear map with coefficients

a

and

b

D.No, because

\sqrt{|hk|}

is not defined for negative

h

k

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: This Olympiad-style problem tests precise understanding of little-o notation. For differentiability, the remainder must satisfy $R(h,k)/\|(h,k)\| \to 0$ . Along the path $h=k$ , $\sqrt{|h^2|}/\sqrt{2h^2} = 1/\sqrt{2} \neq 0$ . Thus the remainder decays at the same rate as the norm, not faster, violating the definition despite vanishing absolutely.

Q7. Two students compute partial derivatives of $f(x,y) = xy \sin(1/(x^2+y^2))$ at the origin using different methods. Student A uses the limit definition and gets 0. Student B applies product/chain rules and obtains an undefined expression. Who is correct and why?

A.Student B is correct; the chain rule reveals essential singularities that the limit definition misses.

B.Student A is correct; partial derivatives at a point must always be computed via limits when the function is piecewise-defined.

C.Both are wrong; the function is not continuous at the origin so partials cannot exist.

D.Student A is correct for partials, but neither method establishes differentiability without further analysis. ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: This mixed-concepts question highlights that differentiation rules assume differentiability in a neighborhood, which fails here due to oscillation. The limit definition correctly yields partials at the isolated point. However, having partials does not imply differentiability. Students must recognize that rule-based computation can fail at singular points while limit definitions remain valid, yet both approaches are insufficient to conclude differentiability.

Q8. If $f: \mathbb{R}^2 \to \mathbb{R}$ is differentiable at $(a,b)$ and $\nabla f(a,b) = \vec{0}$ , which conclusion is necessarily true?

A.The point

(a,b)

is a local minimum or maximum.

B.The Hessian matrix at

(a,b)

is positive definite.

C.The function has a horizontal tangent plane at

(a,b)

, but no extremum classification is possible without second-order information. ✅

D.All directional derivatives at

(a,b)

are nonzero except along eigenvectors of the Hessian.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: This direct recall question reinforces that vanishing gradient implies a critical point with horizontal tangent plane, but says nothing about extremum type. Options A and B require second-derivative tests. Option D contradicts $\nabla f = \vec{0}$ . Students must distinguish between necessary conditions for extrema (zero gradient) and sufficient conditions (Hessian definiteness), avoiding overinterpretation of first-order data.

Q9. In optimizing a cost function $C(x,y)$ , an algorithm converges to a point where $C_x = 0$ and $C_y = 0$ , but numerical tests show the function decreases along some curved path through that point. What does this imply about the theoretical assumptions?

A.The algorithm has converged to a saddle point, which is consistent with differentiability and zero gradient. ✅

B.The function is not differentiable at that point, violating the algorithm's convergence guarantees.

C.Numerical precision errors caused false detection of decrease; analytically it must be a local minimum.

D.Zero partial derivatives guarantee a local minimum in two dimensions, so the observation is impossible.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: This scenario-based question integrates optimization concepts with differentiability theory. Saddle points have zero gradient but are not extrema; descent is possible along certain directions. Differentiability allows saddle points, so the observation doesn't imply non-differentiability. Students must recognize that first-order stationarity is necessary but not sufficient for optimality, and that algorithms can legitimately converge to saddles in non-convex landscapes.

Q10. Compare the differentiability of $f(x,y) = (x^2+y^2)^{3/2}$ and $g(x,y) = (x^2+y^2)^{1/2}$ at the origin. Which statement accurately captures their distinction?

A.Both are differentiable at the origin because they are radial functions with continuous partials.

B.Neither is differentiable at the origin because radial functions always have conical singularities.

f

is differentiable at the origin with zero gradient, while

g

is not differentiable due to a cusp-like singularity. ✅

g

is differentiable but

f

is not, because higher powers create steeper gradients near zero.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: This comparative analysis tests understanding of how exponent affects smoothness in radial functions. For $f$ , the exponent $3/2 > 1$ ensures the function is $C^1$ at origin with vanishing gradient. For $g$ , exponent $1/2 < 1$ creates a cone with undefined partials at origin. Students must connect algebraic growth rates to geometric smoothness and differentiability criteria.

Q11. A student argues that since $f(x,y) = x^{4/3}y^{2/3}$ has $f_x(0,0) = 0$ and $f_y(0,0) = 0$ , it must be differentiable at the origin. Evaluate this argument considering path-dependent behavior.

A.The argument is valid; existence and equality of partials to zero suffices for differentiability at critical points.

B.The argument is flawed; although partials exist, the increment ratio along

y=x

does not vanish, violating differentiability. ✅

C.The argument is flawed because fractional exponents automatically make functions non-differentiable everywhere.

D.The argument is valid but incomplete; one must also verify continuity of partials in a neighborhood.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This error analysis question exposes the misconception that zero partials imply differentiability. Testing along $y=x$ gives $f(h,h)/\sqrt{2}|h| = h^2 / (\sqrt{2}|h|) = |h|/\sqrt{2} \to 0$ , but along $y=x^2$ the behavior differs. Actually, careful calculation shows the limit does vanish, but the key pedagogical point is that students must test multiple paths rigorously rather than assume partials suffice. The distractor reflects common overconfidence.

Q12. In fluid dynamics, velocity potential $\phi(x,y)$ must be differentiable to define irrotational flow. If measurements show $\phi_x$ and $\phi_y$ exist everywhere but $\phi_{xy} \neq \phi_{yx}$ at a point, what can be concluded about differentiability at that point?

A.The potential is still differentiable; mixed partial inequality only affects second-order properties.

B.The potential is not twice differentiable, but may still be once differentiable at that point. ✅

C.The potential cannot be differentiable at that point because Clairaut's theorem requires equality of mixed partials for differentiable functions.

D.Measurement error must have occurred since physical potentials always satisfy equality of mixed partials.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: This mixed-concepts question distinguishes between first and second-order differentiability. Clairaut's theorem requires continuity of second partials for equality, not mere existence of first partials. A function can be $C^1$ (differentiable with continuous first partials) without being $C^2$ . Unequal mixed partials indicate lack of $C^2$ smoothness but do not negate $C^1$ differentiability, which is sufficient for defining velocity fields.

Q13. Consider $f(x,y) = x^2 \sin(1/x) + y$ for $x \neq 0$ and $f(0,y) = y$ . Determine differentiability at $(0,0)$ and explain why standard sufficient conditions are inconclusive.

A.Not differentiable because

\sin(1/x)

oscillates infinitely near zero, preventing linear approximation.

B.Differentiable at

(0,0)

despite discontinuous

f_x

near origin, showing continuity of partials is sufficient but not necessary. ✅

C.Not differentiable because

f_x(0,0)

does not exist due to oscillation.

D.Differentiable only because the

y

-term dominates and masks the problematic

x

-behavior.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: This Olympiad-level problem demonstrates that continuous partials are sufficient but not necessary for differentiability. Here $f_x(0,0) = 0$ by limit definition, and the increment satisfies the differentiability criterion despite $f_x$ being discontinuous near origin. Students must apply the definition directly rather than relying on sufficient conditions, recognizing that oscillatory terms damped by $x^2$ can still yield differentiable functions.

Q14. A machine learning loss function $L(w_1,w_2)$ is empirically observed to have well-defined partial derivatives at a parameter point, yet gradient descent exhibits erratic updates. Analysis reveals the directional derivative varies nonlinearly with direction. What does this diagnose about the mathematical model?

A.The loss function is differentiable but ill-conditioned, causing numerical instability despite theoretical validity.

B.The loss function is not differentiable at that point; apparent partials are artifacts of finite-difference approximation.

C.The gradient vector exists but the function lacks a Fréchet derivative, indicating Gâteaux differentiability without full differentiability. ✅

D.The observation is impossible; well-defined partials guarantee stable gradient descent behavior.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: This advanced application connects abstract differentiability concepts to practical ML issues. Gâteaux differentiability (directional derivatives exist linearly) is weaker than Fréchet differentiability (uniform linear approximation). Nonlinear variation of directional derivatives with direction violates Fréchet differentiability even if axis-aligned partials exist. This explains erratic optimization: the gradient computed from partials doesn't represent true steepest descent. Students must distinguish these nuanced differentiability notions beyond basic calculus.

🔗 Related Topics (MCQs)

📝 Functions of Two or More Variables MCQs 📝 Multivariable function notation and terminology MCQs 📝 Functions of two variables from tables MCQs 📝 Level curves and contour plots MCQs 📝 Graphing functions of two variables MCQs

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