Question

The number of critical points of the function $f(x) = (x - 2)^{2/3}(2x + 1)$ is:

• A. 2
• B. 0
• C. 1
• D. 3

Answer 1

The Correct Option is A

Solution:

The given function is \(f(x)\). Its derivative is:

\[ f'(x) = \frac{2}{3}(x - 2)^{-1/3}(2x + 1) + (x - 2)^{2/3}(2). \]

Simplify the numerator:

\[ f'(x) = \frac{2}{3} \cdot \frac{(2x + 1) + 3(x - 2)}{(x - 2)^{1/3}}. \]

Expand and simplify:

\[ (2x + 1) + 3(x - 2) = 5x - 5. \]

Thus:

\[ f'(x) = \frac{2(5x - 5)}{3(x - 2)^{1/3}}. \]

Critical points:

1. Set \(f'(x) = 0\):
2. The derivative \(f'(x)\) is undefined at \(x = 2\).

Hence, the critical points are:

\[ x = 1 \quad \text{and} \quad x = 2. \]

Final Answer: 2.