Question

If \( f(x) = x^5 + 2x - 3 \), then \( (f^{-1})'(-3) = \)

• A. 0
• B. -3
• C. 1
• D. 1/2

Hint:

For inverse function derivative, use \( (f^{-1})'(y) = \frac{1}{f'(x)} \) where \( y = f(x) \).

Answer 1

The Correct Option is D

Find \( x \) such that \( f(x) = -3 \): 
\[ x^5 + 2x - 3 = -3 \Rightarrow x^5 + 2x = 0 \Rightarrow x (x^4 + 2) = 0 \Rightarrow x = 0. \] Check: \( f(0) = 0^5 + 2 \cdot 0 - 3 = -3 \). So, \( f^{-1}(-3) = 0 \). 
Derivative of inverse: \( (f^{-1})'(y) = \frac{1}{f'(x)} \) where \( y = f(x) \). 
\[ f'(x) = 5x^4 + 2, f'(0) = 5 \cdot 0 + 2 = 2. \] \[ (f^{-1})'(-3) = \frac{1}{f'(0)} = \frac{1}{2}. \] Answer: \( \frac{1}{2} \).