Question

The function f(x) = 2x + 3(x)2/3, x ∈ 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 two points of local maxima and exactly one point of local minima
• D. exactly one point of local maxima and exactly one point of local minima

Answer 1

Answer: (D)

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

Answer 2

The function is $f(x) = 2x + 3x^{2/3}$. The first derivative is $f'(x) = 2 + 2x^{-1/3} = 2 \left(1 + \frac{1}{x^{1/3}}\right)$. Critical points occur where $f'(x) = 0$ or $f'(x)$ is undefined. $f'(x)$ is undefined at $x=0$. $f'(x) = 0 \implies 1 + \frac{1}{x^{1/3}} = 0 \implies x^{1/3} = -1 \implies x = -1$. The critical points are $x=-1$ and $x=0$.

Analyzing the sign of $f'(x) = 2 \left(\frac{x^{1/3} + 1}{x^{1/3}}\right)$:

• For $x < -1$, $f'(x) > 0$ (increasing).
• For $-1 < x < 0$, $f'(x) < 0$ (decreasing).
• For $x > 0$, $f'(x) > 0$ (increasing).

At $x=-1$, $f(x)$ changes from increasing to decreasing (local maximum). At $x=0$, $f(x)$ changes from decreasing to increasing (local minimum). Therefore, there is exactly one point of local maxima and exactly one point of local minima.