Question

The function $f(x) = 2x + 3(x)^{\frac{2}{3}}, x \in \mathbb{R}$, has

• A. exactly one point of local minima and no point of local maxima
• B. exactly one point of local maxima and no point of local minima
• C. exactly one point of local maxima and exactly one point of local minima
• D. exactly two points of local maxima and exactly one point of local minima

Answer 1

Answer: (C)

exactly one point of local maxima and exactly one point of local minima

Answer 2

Solution: To find the local maxima and minima, we calculate the derivative $f'(x)$ and analyze the critical points.

Step 1. Finding $f'(x)$:
$f(x) = 2x + 3(x)^{\frac{1}{3}}$
$f'(x) = 2 + 2x^{-\frac{2}{3}}$
$= 2 \left( 1 + \frac{1}{x^{\frac{2}{3}}} \right)$

Step 2. Setting $f'(x) = 0$ to find critical points:
$2 \left( 1 + \frac{1}{x^{\frac{2}{3}}} \right) = 0$
$1 + \frac{1}{x^{\frac{2}{3}}} = 0$
$x^{\frac{2}{3}} = -1 \implies x = -1$

Step 3. Analyzing the sign of $f'(x)$ around the critical points $x = -1$ and $x = 0$:

So, the function has a local maximum (M) at $x = -1$ and a local minimum (m) at $x = 0$.

The Correct Answer is: Exactly one point of local maxima and exactly one point of local minima