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Question

Which of the following function has a point of inflection at x = 0 ?


A

f(x)=sin(x)

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B

f(x)=cos(x)

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C

f(x)=x2

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D

f(x)=x3

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Solution

The correct option is D

f(x)=x3


We will find f”(x) and substitute x=0 to find check if zero is a point of inflection. At point of inflection, we also have f(x) 0

Let’s check each option one by one.

  1. f(x) = sin (x) (Given)

f’(x) = cos(x)

f”(x) = - sin (x)

f”(0) = 0

So x = 0 could be a point of inflection.

To confirm it let’s check f”’(x) as well.

”’(x) = - cos(x)

f(0)0

Hence, we can say x = 0 is a point of inflection for f(x) = sin (x).

b. f(x) = cos(x) (Given)

f’(x) = - sin(x)

f”(x) = - cos(x)

f”(0) = -1

Since, f”(0) ≠ 0 , x = 0 is not a point of inflection.

c. f(x)=x2

f’’(x) = 2

=> f”(0) = 2

Since f”(0) is not equal to zero, x = 0 is not a point of inflection

d. f(x)=x3

f”(x) = 6x

=> f”(0) = 0

This means x = 0 could be a point of inflection. To confirm this, we will find f’”(x) at x = 0

f”’(x) = 6

=> f’”(0) = 6

Since f(0)0 and f”(0) =0, we can say x =0 is a point of inflection.

So the options a and d are correct


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