Let f(x)=x⁵+2 e^(x / 4) for all x ∈ R. Consider a function g(x) such that (g ° f)(x)=x for all x ∈ R. Then the value of 8 g′(2) is :

Solution

Maths sir · 5 steps

Do not hunt for g. Find which x gives f(x) = 2, then flip the slope of f there.

1. g undoes f. f'(x) = 5x⁴ + 1/2e^(x/4) > 0 for every x, so f is one-one. Since g(f(x)) = x, g is the inverse of f.
2. Differentiate. Differentiate g(f(x)) = x with the chain rule: g'(f(x)) · f'(x) = 1.

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

3. Find the right x. You need f(x) = 2. Try x = 0: f(0) = 0 + 2e⁰ = 2. So g'(2) = 1/f'(0).
4. Slope at zero. f'(0) = 5(0)⁴ + 1/2e⁰ = 1/2, so g'(2) = 1/(1/2) = 2.
5. So. 8g'(2) = 8 × 2 = 16, which is option (c).

The idea to keep. For an inverse, g'(b) = 1/f'(a) where f(a) = b. The real work is spotting a, and x = 0 is the first value to try.

Answer from the NTA final answer key. Solution by PrepWiser.