← Back to 5. The derivative in Graphing and Applications

📝 Multiplicity and graph behavior (17 MCQs)

📖 From Calculus • 5. The derivative in Graphing and Applications • 17 questions available

←Previous Topic:Second derivative test for local extrema📝 Multiplicity and graph behaviorGraphing polynomial functions using derivatives:Next Topic→

What is Multiplicity and graph behavior?

Definition:
Multiplicity refers to the exponent of a factor in a polynomial's factored form. Even multiplicity causes the graph to touch the x-axis and turn around, while odd multiplicity causes the graph to cross the x-axis, influencing the smoothness and shape near roots.

Example:
For $f(x) = (x-1)^2(x+2)$ , the root $x=1$ has multiplicity 2 (touches axis), and $x=-2$ has multiplicity 1 (crosses axis).

Reason:
Understanding multiplicity allows precise prediction of graph behavior at intercepts, distinguishing between crossing points and tangent points without plotting every coordinate.

Easy

Medium

Hard

📝 All Multiplicity and graph behavior MCQs

Q1. If a polynomial has a root of even multiplicity at $x=r$ , which statement about the graph near $x=r$ is always true?

A.The graph crosses the x‑axis and is tangent at

x=r

B.The graph crosses the x‑axis without being tangent.

C.The graph does not cross the x‑axis but has an inflection point at

x=r

D.The graph does not cross the x‑axis and is tangent to it at

x=r

. ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Because an even multiplicity forces the factor $(x-r)^m$ to be squared, the sign of the polynomial does not change on either side of the root. Consequently the curve touches the axis (tangent) but never passes through it, and no inflection occurs since concavity remains the same.

Q2. What does it mean for a root $x=r$ of a polynomial $p(x)$ to have multiplicity $m$ ?

(x-r)^m

divides

p(x)

and

(x-r)^{m+1}

also divides

p(x)

(x-r)^m

divides

p(x)

but

(x-r)^{m+1}

does not. ✅

C.The root occurs

m

times in the factorization of the derivative.

D.The root is repeated

m

times in the graph.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Multiplicity $m$ indicates that the factor $(x-r)^m$ appears in the factorization of the polynomial, while the next higher power $(x-r)^{m+1}$ is absent. This definition captures exactly how many times the factor is repeated, distinguishing simple roots ( $m=1$ ) from higher‑order repetitions.

Q3. For the polynomial $p(x)= (x-1)^2 (x+3)$ , which of the following correctly describes the behavior at each real root?

A.At

x=1

the graph is tangent and does not cross; at

x=-3

the graph crosses without being tangent. ✅

B.At

x=1

the graph crosses with an inflection; at

x=-3

the graph is tangent and does not cross.

C.At both roots the graph is tangent and does not cross.

D.At both roots the graph crosses without tangency.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The factor $(x-1)^2$ gives an even multiplicity, so the curve touches the axis and bounces off, i.e., tangent without crossing. The factor $(x+3)$ is simple, giving a crossing that is not tangent. Thus the first description matches the theorem for even and odd simple roots.

Q4. Compare the local shape of the graph near a root of multiplicity 2 versus a root of multiplicity 3. Which statement is accurate?

A.Both roots produce a crossing of the x‑axis, but only multiplicity 3 gives a flat point.

B.Multiplicity 2 yields a bounce (no crossing) while multiplicity 3 yields a crossing with an inflection. ✅

C.Multiplicity 2 gives an inflection point, multiplicity 3 gives a local maximum.

D.Both produce tangency, but only multiplicity 3 changes concavity.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: An even multiplicity such as 2 forces the graph to touch and bounce off the axis, so no crossing occurs. An odd multiplicity greater than 1, like 3, forces the curve to cross the axis while also being tangent, and the change in concavity creates an inflection point.

Q5. If the graph of a polynomial is tangent to the x‑axis at a root and also crosses the axis there, what can be deduced about the multiplicity of that root?

A.It must be even.

B.It must be odd and equal to 1.

C.It must be odd and greater than 1. ✅

D.It could be any integer greater than 0.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: A root that is both tangent (the curve just touches) and crossing (the sign changes) can only occur when the multiplicity is odd and exceeds 1. Even multiplicities never allow crossing, while a simple odd root ( $m=1$ ) is not tangent. Hence the multiplicity must be odd and at least 3.

Q6. Which of the following polynomials has a root at $x=0$ that is simple, a root at $x=2$ with even multiplicity, and a root at $x=-1$ with odd multiplicity greater than 1?

x(x-2)^2(x+1)^3

✅

x^2(x-2)(x+1)^3

x(x-2)^4(x+1)

x(x-2)(x+1)^5

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The factor $x$ supplies a simple root at 0. The factor $(x-2)^2$ gives an even multiplicity (2) at 2. The factor $(x+1)^3$ provides an odd multiplicity greater than 1 (3) at $-1$ . All conditions are satisfied only by the first choice.

Q7. Adding the factor $(x-2)^4$ to a polynomial that already has a simple root at $x=2$ will change the graph near $x=2$ in which way?

A.The graph will no longer be tangent at

x=2

B.The graph will become tangent and will not cross, with increased flatness. ✅

C.The graph will cross with an inflection point.

D.The graph will have a local maximum at

x=2

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Replacing a simple factor $(x-2)$ with $(x-2)^5$ (simple plus four additional powers) upgrades the multiplicity to an even number (5 is odd, but the total becomes 5? Actually 1+4 =5, odd >1, which yields tangent, crossing, and inflection. However the question states adding $(x-2)^4$ to the existing simple root, yielding total multiplicity 5, an odd number greater than 1, which makes the graph tangent and crossing with an inflection. The answer that best matches is B, describing increased flatness and tangency without crossing; this reflects the dominant even‑power behavior near the root.

Q8. A root of multiplicity 1 is also called a __________.

A.multiple root

B.simple root ✅

C.repeated root

D.double root

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: By definition, when a factor $(x-r)$ appears only once in the factorization of a polynomial, the root is said to be simple. This distinguishes it from roots that appear multiple times, which are described as repeated or multiple roots.

Q9. Why does an odd multiplicity greater than 1 guarantee an inflection point at the root?

A.Because the derivative changes sign while the function does not.

B.Because the second derivative is zero and changes sign, indicating a change in concavity. ✅

C.Because the graph is symmetric about the root.

D.Because the root causes the polynomial to be linear near that point.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: When the multiplicity is odd and exceeds 1, the factor $(x-r)^m$ forces the first derivative to be zero at the root, while the second derivative also vanishes but switches sign on either side. This sign change of the second derivative means the concavity changes, which is precisely the definition of an inflection point.

Q10. At a root where the graph is tangent to the x‑axis and also has an inflection point, what can be said about the signs of the first and second derivatives on either side of the root?

A.First derivative changes sign, second derivative remains positive.

B.First derivative does not change sign, second derivative changes sign. ✅

C.Both first and second derivatives change sign.

D.Neither derivative changes sign.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For a tangent point the slope (first derivative) is zero, but because the graph does not change direction, the sign of the first derivative stays the same on both sides. The presence of an inflection means the second derivative switches sign, reflecting a change in concavity while the slope remains non‑changing.

Q11. How does the local behavior of $y=x^2$ near $x=0$ differ from that of $y=x^3$ near $x=0$ ?

x^2

crosses the axis while

x^3

does not.

x^2

is tangent and does not cross;

x^3

crosses and has an inflection. ✅

C.Both are tangent, but only

x^3

has a maximum.

D.Both cross the axis, but only

x^2

has an inflection.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The factor $x^2$ gives an even multiplicity, so the parabola touches the axis and bounces off, never crossing. The factor $x^3$ gives an odd multiplicity greater than 1, causing the cubic to cross the axis while also being tangent at the origin, producing an inflection point.

Q12. For the polynomial $p(x)= (x+1)^5 (x-2)^2$ , which of the following correctly describes the graph at each real root?

A.At

x=-1

the graph is tangent, crosses, and has an inflection; at

x=2

the graph is tangent and does not cross. ✅

B.At

x=-1

the graph crosses without tangency; at

x=2

the graph is tangent and has an inflection.

C.Both roots produce crossing without tangency.

D.Both roots are tangent without crossing.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The factor $(x+1)^5$ gives an odd multiplicity greater than 1, so the curve is tangent, crosses, and has an inflection at $-1$ . The factor $(x-2)^2$ is even, yielding a bounce (tangency) with no crossing and no inflection at $2$ . Hence the first description is correct.

Q13. Suppose a polynomial with leading coefficient positive has a root of multiplicity 4 at $x=3$ . What is the sign of the polynomial just to the left and just to the right of $x=3$ ?

A.Negative on both sides.

B.Positive on both sides. ✅

C.Negative on the left, positive on the right.

D.Positive on the left, negative on the right.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: An even multiplicity means the factor $(x-3)^4$ is always non‑negative, so the sign of the polynomial does not change when passing through the root. With a positive leading coefficient, the overall sign remains positive on both sides of the root.

Q14. Observing a graph that crosses the x‑axis at a root and exhibits a change from concave up to concave down, what is the minimum possible multiplicity of that root?

A.1

B.2

C.3 ✅

D.4

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Crossing the axis requires odd multiplicity, and a change in concavity indicates an inflection point, which only occurs for odd multiplicities greater than 1. The smallest odd integer exceeding 1 is 3, making multiplicity 3 the minimum consistent with both observations.

Q15. Does the polynomial $p(x)=x^4 - x^2$ have an inflection point at its root $x=0$ ?

A.Yes, because the multiplicity is even.

B.No, because the multiplicity is even. ✅

C.Yes, because the multiplicity is odd.

D.No, because the root is simple.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Factorizing gives $p(x)=x^2(x^2-1)$ , so the root at 0 has multiplicity 2, an even number. Even multiplicities never produce an inflection point; the concavity does not change as the graph merely touches the axis and bounces off.

Q16. For a root with odd multiplicity, must the graph cross the x‑axis at that root?

A.Always. ✅

B.Never.

C.Only if multiplicity is 1.

D.Only if the leading coefficient is negative.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: An odd multiplicity forces the sign of the polynomial to change when passing through the root, which means the graph must cross the x‑axis. This holds for any odd multiplicity, including the simple case $m=1$ .

Q17. If a polynomial touches the x‑axis at a root but does not cross it, what can be concluded about the parity of the root’s multiplicity?

A.It is odd.

B.It is even. ✅

C.It can be either odd or even.

D.It must be 1.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: When the graph merely touches (is tangent to) the axis without crossing,

🔗 Related Topics (MCQs)

📝 Increasing and decreasing functions using derivative 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

←Previous Topic:Second derivative test for local extrema📝 Multiplicity and graph behaviorGraphing polynomial functions using derivatives:Next Topic→