Let f ( x ) = x 5 + 2 e x / 4 for all x ? R . Consider a function g ( x ) such that ( g o f ) ( x ) = x for all x ? R . Then the value of 8 g ' ( 2 ) is [2024]

Question

Let $f (x) = x^{5} + 2 e^{x / 4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g o f)$ $(x) = x$ for all $x \in R$ . Then the value of $8 g' (2)$ is [2024]

Smart Achievers · Accepted Answer

(2)

We have, $(g o f) (x) = x \Rightarrow g (f (x)) = x$

Differentiating w.r.t. $x$ , we get

$g' (f (x)) \cdot f' (x) = 1 \Rightarrow g' (f (x)) = \frac{1}{f' (x)}$

Now, $f (x) = x^{5} + 2 e^{x / 4} \Rightarrow f' (x) = 5 x^{4} + \frac{1}{2} e^{x / 4}$

$∴$ $f' (0) = \frac{1}{2}, f (0) = 2 \Rightarrow g' (2) = 2$

Hence, $8 g' (2) = 16$

Solution

Q.
Let $f (x) = x^{5} + 2 e^{x / 4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g o f)$ $(x) = x$ for all $x \in R$ . Then the value of $8 g' (2)$ is [2024]

1 4

2 16

3 8

4 2

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Solution

Q. Let f(x)=x5+2ex/4 for all x∈R. Consider a function g(x) such that (gof) (x)=x for all x∈R. Then the value of 8g'(2) is [2024]

1 4

2 16

3 8

4 2

Ans. (2) We have, (gof)(x)=x⇒g(f(x))=x Differentiating w.r.t. x, we get g'(f(x))·f'(x)=1⇒g'(f(x))=1f'(x) Now, f(x)=x5+2ex/4⇒f'(x)=5x4+12ex/4 ∴ f'(0)=12, f(0)=2⇒g'(2)=2 Hence, 8g'(2)=16

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Q.
Let $f (x) = x^{5} + 2 e^{x / 4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g o f)$ $(x) = x$ for all $x \in R$ . Then the value of $8 g' (2)$ is [2024]