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

The Correct Option is C

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