Question 74

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

g is the inverse of f: g(f(x)) = x. Differentiating: g'(f(x))·f'(x) = 1

g'(f(x)) = 1/f'(x)

f(x) = x⁵ + 2e^(x/4). f(0) = 0 + 2 = 2.

So g'(2) = g'(f(0)) = 1/f'(0)

f'(x) = 5x⁴ + (1/2)e^(x/4)

f'(0) = 0 + 1/2 = 1/2

g'(2) = 1/(1/2) = 2

8g'(2) = 16

The correct answer is Option 4: 16.

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