JEE Main 2020 · Mathematics · Differentiation and Application of Derivatives · medium

Let f(x)=x^3+2x-3 and let g be the inverse function of f. Then g''(0) equals

Single correct option

Let

f(x)=x^3+2x-3

and let

g

be the inverse function of

f

. Then

g''(0)

equals

Show answer and solution

Answer: (D) $-\dfrac{6}{125}$

First find the point where

f(x)=0

f(1)=1+2-3=0

, so

g(0)=1

. Since

g(f(x))=x

, differentiating gives

g'(f(x))\,f'(x)=1

, so

g'(y)=\dfrac1{f'(g(y))}

. Differentiating again with respect to

y

g''(y)=-\frac{f''(g(y))\,g'(y)}{f'(g(y))^2}=-\frac{f''(x)}{f'(x)^3},\quad x=g(y).

Here

f'(x)=3x^2+2

and

f''(x)=6x

, so at

x=1

f'=5

f''=6

. Thus

g''(0)=-\dfrac{6}{125}

. The distractor

-\dfrac{6}{25}

comes from forgetting one factor of

f'

in the denominator.

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