The inverse ?

Calculus Level 3

If the function f ( x ) = x 3 + e x 2 \displaystyle f(x)=x^{3}+e^{\frac{x}{2}} and g ( x ) = f − 1 ( x ) \displaystyle g(x) =f^{-1}(x) then the value of g ′ ( 1 ) \displaystyle g'(1) is ?


The answer is 2.

Keshav Tiwari by Keshav Tiwari , 23, India

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2 solutions

Prakhar Gupta
Feb 5, 2015

From definition of inverse function we can write that:- g ( f ( x ) ) = x g(f(x)) = x Differentiating both sides we get:- g ′ ( f ( x ) ) f ′ ( x ) = 1 g'(f(x)) f'(x) =1 ⟹ g ′ ( f ( x ) ) = 1 f ′ ( x ) \implies g'(f(x)) = \dfrac{1}{f'(x)} Put x = 0. g ′ ( f ( 0 ) ) = 1 f ′ ( 0 ) g'(f(0)) = \dfrac{1}{f'(0)} Hence we get:- g ′ ( 1 ) = 2 \boxed{g'(1) = 2}

6 Helpful 0 Interesting 0 Brilliant 0 Confused
Mahimn Bhatt
Jan 19, 2015

g(f(x))=x

Differentiate both the sides.

Put x=0 ,

get the value.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Yes , that's the easy way ! This is a previous IIT question . :)

Keshav Tiwari - 6 years, 4 months ago

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Which year?

Jee mains or advance?

Mahimn Bhatt - 6 years, 4 months ago

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Integer type 2009.

Keshav Tiwari - 6 years, 4 months ago

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@Keshav Tiwari – It was hell easy...

Mahimn Bhatt - 6 years, 4 months ago

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@Mahimn Bhatt – Try this https://brilliant.org/problems/aod-maximum/?group=Lir31zAVzjF0&ref_id=578097

Keshav Tiwari - 6 years, 4 months ago

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