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📝 Increasing and decreasing functions using derivative (21 MCQs)

📖 From Calculus • 5. The derivative in Graphing and Applications • 21 questions available

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What is Increasing and decreasing functions using derivative?

Definition:
A function $f(x)$ is increasing on an interval if $f'(x) > 0$ and decreasing if $f'(x) < 0$ . This derivative test determines monotonicity by analyzing the sign of the first derivative across specific domains to understand rate of change behavior.

Example:
For $f(x) = x^2$ , $f'(x) = 2x$ . Since $2x < 0$ for $x < 0$ , the function decreases on $(-\infty, 0)$ .

Reason:
The sign of the slope indicates direction; positive slope means rising values while negative slope means falling values as $x$ increases.

Easy

Medium

Hard

📝 All Increasing and decreasing functions using derivative MCQs

Q1. For the function $f(x)=x^2$ , on which interval is the function increasing?

(-∞,0)

(0,∞)

✅

(-∞,∞)

D.None of the above

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Because the derivative $f'(x)=2x$ is positive when $x>0$ . Thus the function rises as we move rightward for all x greater than zero, making $(0,∞)$ the interval of increase.

Q2. If a continuous function $g$ satisfies $g'(x)>0$ for every $x$ in $(1,4)$ and $g(1)=3$ , which of the following must be true?

g(2) < g(1)

g(2) > g(1)

✅

g(4) = g(1)

g

is decreasing on

(1,4)

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A positive derivative on an interval implies the function is strictly increasing there. Therefore any point with a larger x‑value must have a larger function value, guaranteeing $g(2) > g(1)$ .

Q3. Compare the intervals of increase for $h(x)=\sin x$ and $k(x)=\cos x$ on $[0,2\pi]$ . Which statement is correct?

A.Both increase on

(0,\pi/2)

h

increases on

(0,\pi)

while

k

increases on

(\pi,2\pi)

h

increases on

(0,\pi)

and

k

increases on

(0,\pi/2)

✅

D.Neither function increases on any subinterval

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: $\sin x$ has derivative $\cos x>0$ on $(0,\pi)$ , so it increases there. $\cos x$ has derivative $-\sin x>0$ only on $(0,\pi/2)$ . Hence the third option correctly describes their increasing intervals.

Q4. A function $p$ is said to be increasing on an interval I. Which of the following statements must be false?

A.There exists

a<b

in I with

p(a)=p(b)

B.The derivative

p&#039;(x)

is negative for some x in I ✅

C.Both A and B

D.None of the above

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: If a function is increasing on an interval, its derivative cannot be negative at any point (except possibly where it is zero). Thus the claim that the derivative is negative somewhere must be false.

Q5. Suppose $q$ is differentiable on $[−2,2]$ and $q'(x)=0$ only at $x=0$ . Which interval must contain a local maximum of $q$ ?

(-2,0)

(0,2)

C.Both intervals

D.Neither interval ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Since the derivative is non‑zero everywhere except at 0, we cannot guarantee a sign change that would produce a local extremum. Hence no interval can be asserted to contain a guaranteed local maximum.

Q6. Given two functions $f$ and $g$ where $f$ is increasing on $[a,b]$ and $g$ is decreasing on the same interval, what can be said about the monotonicity of the product $h(x)=f(x)g(x)$ assuming all values are positive?

h

is increasing

h

is decreasing

h

may be increasing or decreasing depending on magnitudes ✅

h

is constant

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: When both factors are positive, the increasing trend of $f$ tends to raise the product while the decreasing trend of $g$ tends to lower it. The net effect depends on which factor changes more rapidly, so the product’s monotonicity cannot be predetermined.

Q7. Let $r$ be a twice‑differentiable function on $\mathbb{R}$ with $r'(x)>0$ for all $x$ and $r''(x)<0$ for all $x$ . Which of the following best describes the graph of $r$ ?

A.Increasing and concave up

B.Increasing and concave down ✅

C.Decreasing and concave up

D.Decreasing and concave down

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A positive first derivative means the function rises as $x$ increases, while a negative second derivative indicates the graph bends downward. Hence the function is increasing and concave down throughout its domain.

Q8. If a function $s$ is constant on an interval, which of the following must hold for its derivative on that interval?

A.Derivative is positive

B.Derivative is negative

C.Derivative is zero ✅

D.Derivative does not exist

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: A constant function has the same output for every input, so its rate of change is zero. By definition, the derivative of a constant function on that interval is identically zero.

Q9. Two functions $f(x)=e^{x}$ and $g(x)=\ln x$ (for $x>0$ ) are compared. Which statement correctly reflects their monotonic behavior on their domains?

A.Both are increasing ✅

B.Both are decreasing

f

is increasing,

g

is decreasing

f

is decreasing,

g

is increasing

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The derivative of $e^{x}$ is $e^{x}>0$ for all $x$ ; the derivative of $\ln x$ is $1/x>0$ for $x>0$ . Both derivatives are positive, so each function increases throughout its domain.

Q10. A function $t$ satisfies $t'(x)=\frac{1}{1+x^{2}}$ . Which of the following statements is true about the intervals where $t$ is increasing?

t

is increasing on all real numbers ✅

t

is increasing only on

(-∞,0)

t

is increasing only on

(0,∞)

t

is not increasing anywhere

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The derivative $\frac{1}{1+x^{2}}$ is always positive regardless of the sign of $x$ . A positive derivative everywhere guarantees that the function is increasing on the entire real line.

Q11. Let $u$ be differentiable on $[0,3]$ with $u'(0)=2$ and $u'(3)=-1$ . Which of the following must be true according to the Mean Value Theorem?

A.There exists

c\in(0,3)

with \(u'(c)=0

B.There exists

c\in(0,3)

with \(u'(c)=\frac{u(3)-u(0)}{3}

u

is decreasing on the whole interval ✅

u

is increasing on the whole interval

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The Mean Value Theorem guarantees a point $c$ where the instantaneous rate of change equals the average rate of change over the interval, i.e., $u'(c)=\frac{u(3)-u(0)}{3}$ .

Q12. Consider $v(x)=x^{3}-3x$ . On which intervals is $v$ decreasing?

(-∞,-1)

and

(1,∞)

(-1,1)

✅

(-∞,0)

and

(0,∞)

D.Nowhere

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: $v'(x)=3x^{2}-3=3(x^{2}-1)$ . The derivative is negative when $-1<x<1$ , so the function decreases exactly on the open interval $(-1,1)$ .

Q13. Which of the following best describes a function that is increasing on $[a,b]$ and decreasing on $[b,c]$ ?

A.It has a local maximum at

b

✅

B.It has a local minimum at

b

C.It is constant at

b

D.It is undefined at

b

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: If the function rises up to $b$ and then falls after $b$ , the point $b$ is a peak, i.e., a local maximum.

Q14. If $f$ is increasing and $g$ is increasing, what can be said about the composition $h(x)=f(g(x))$ on an interval where both are defined?

h

is decreasing

h

is increasing ✅

h

is constant

D.No conclusion can be drawn

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: An increasing outer function applied to an increasing inner function preserves order: larger inputs to $g$ give larger outputs, which $f$ then maps to larger values, so the composition is increasing.

Q15. Given $w(x)=\frac{1}{x}$ for $x>0$ , which interval correctly describes where $w$ is decreasing?

(0,1)

(1,∞)

✅

(-∞,0)

D.No interval

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The derivative $w'(x)=-\frac{1}{x^{2}}$ is negative for all positive $x$ . Hence the function is strictly decreasing on any interval of positive numbers, such as $(1,∞)$ .

Q16. A function $y$ satisfies $y'(x)=\sin x$ . On which intervals is $y$ concave down?

A.Where

\sin x>0

B.Where

\sin x<0

C.Where

\cos x>0

D.Where

\cos x<0

✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Concavity is determined by the sign of the second derivative. Differentiating again gives $y''(x)=\cos x$ . The function is concave down wherever $\cos x<0$ .

Q17. If a function's graph moves upward as you move right, which term best describes its behavior?

A.Decreasing

B.Increasing ✅

C.Constant

D.Oscillating

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: An upward trend from left to right indicates that the function's values grow with larger inputs, which is precisely the definition of an increasing function.

Q18. For $z(x)=\ln(x^2+1)$ , determine the intervals where $z$ is increasing.

A.All real numbers

x>0

✅

x<0

D.No interval

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: $z'(x)=\frac{2x}{x^{2}+1}$ is positive exactly when $x>0$ . Therefore the function increases on the positive half‑line and decreases on the negative half‑line.

Q19. If $f$ is decreasing on an interval and you reflect its graph across the x‑axis, what is the monotonicity of the new function?

A.Increasing ✅

B.Decreasing

C.Constant

D.Cannot determine

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Reflecting across the x‑axis multiplies the function by $-1$ . Changing the sign of a decreasing function reverses its order, turning it into an increasing function.

Q20. Suppose $p$ is continuous on $[a,b]$ and differentiable on $(a,b)$ with $p'(x)\ge 0$ for all $x$ and $p'(c)>0$ for some $c$ . Which statement must be true?

p

is constant on

[a,b]

p

is strictly increasing on

[a,b]

p

is increasing but not necessarily strictly on

[a,b]

✅

p

may be decreasing somewhere

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A non‑negative derivative guarantees the function never decreases. Since the derivative is positive at least at one point, the function is not constant, but it may have flat portions, so it is increasing (non‑decreasing) overall.

Q21. Let $f(x)=x^{4}-4x^{2}$ . Find the intervals where $f$ is increasing.

(-∞,-\sqrt{2})

and

(\sqrt{2},∞)

(-√2,0)

and

(\sqrt{2},∞)

✅

(-√2,0)

and

(0,√2)

D.All real numbers

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: $f'(x)=4x(x^{2}-2)$ . The sign is positive on $(-√2,0)$ and on $(√2,∞)$ ; therefore the function increases on those two intervals.

🔗 Related Topics (MCQs)

📝 Concavity and inflection points calculus MCQs 📝 Logistic growth curves calculus MCQs 📝 Analysis of Functions II (Relative Extrema) in calculus MCQs 📝 Relative maxima and minima local extrema MCQs 📝 First derivative test for local extrema MCQs

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