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

To find the critical points of the function \( f(x) = 2x + 3(x)^{\frac{2}{3}} \), we first calculate its derivative. The function is defined for all real numbers. We will examine its first and second derivatives to identify local maxima and minima.

1. Compute the first derivative of \( f(x) \):

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

1. Set the first derivative to zero to locate critical points:

\(2 + 2(x)^{-\frac{1}{3}} = 0\)

1. Simplify and solve for \( x \):

\((x)^{-\frac{1}{3}} = -1 \Rightarrow x^{-\frac{1}{3}} = -1 \Rightarrow x = (-1)^{-3} = -1\)

1. Determine the nature of the critical points by calculating the second derivative:

\(f''(x) = \frac{d}{dx} [2 + 2(x)^{-\frac{1}{3}}] = 0 - \frac{2}{3}(x)^{-\frac{4}{3}}\)

1. Evaluate the second derivative at \( x = -1 \):

\(f''(-1) = - \frac{2}{3}(-1)^{-\frac{4}{3}} = -\frac{2}{3}(-1)^{4/3} = -\frac{2}{3}(1) = -\frac{2}{3}\)

1. Classify the critical point:

Since \( f''(-1) \lt 0 \), the function has a local maximum at \( x = -1 \).

1. Analyze behavior near \( x = 0 \):
2. Observe the function's behavior as \( x \) approaches \( 0 \) from the positive side, given the negative exponents in the derivative.

\(f'(x) = 2 + \frac{2}{x^{1/3}}\). As \( x \to 0^+ \), \( f'(x) \) becomes very large.

This indicates that the function increases as it approaches \( x = 0 \). Since \(f'\) changes sign around \(x=0\), there is a minimum near \(x=0\).

1. Summary of critical points:
   • A local maximum at \(x = -1\).
   • A local minimum at \(x \rightarrow 0^+\).

The function \( f(x) \) exhibits exactly one local maximum and exactly one local minimum.