Question

If f(x) = x⁵ + 2x – 3, then (f^–1)¹ (–3) = ______.

Options

• 0

• – 3

• `-1/3`

• `1/2`

Answer 1

If f(x) = x⁵ + 2x – 3, then (f^–1)¹ (–3) = `bbunderline(1/2)`.

Explanation:

Given:

f(x) = x⁵ + 2x − 3

We need to find (f⁻¹)′(−3)

Step 1: Formula for derivative of inverse function

(f⁻¹)′(a) = `1/(f'(f^(-1)(a)))`

So we must find the value of x such that f(x) = a = −3.

Step 2: Find x such that f(x) = −3

x⁵ + 2x − 3 = −3

Simplify:

x⁵ + 2x = 0

x(x⁴ + 2) = 0

So x = 0.

Step 3: Compute f′(x)

f′(x)  = 5x⁴ + 2

At x = 0,

f′(0) = 5(0)⁴ + 2 = 2

Step 4: Substitute in the formula

(f⁻¹)′(−3) = `1/(f'(0)) = 1/2`