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

Answer: (A)

2

Answer 2

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.