Question

The maximum value of \(x^{2 / 3}+(x-2)^{2 / 3}\) is

• A. 0
• B. 2
• C. \(2^{\frac{2}{3}}\)
• D. 1

Hint:

Set the first derivative to zero and check the sign change around the critical point.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Let \(f(x) = x^{2/3} + (x-2)^{2/3}\). Find where \(f'(x) = 0\).

Step 2: Derivative
\(f'(x) = \dfrac{2}{3}\left[x^{-1/3} + (x-2)^{-1/3}\right]\). Setting this to zero gives \(x^{-1/3} = -(x-2)^{-1/3}\), so \(x = 2 - x\) and \(x = 1\).

Step 3: Sign check
For \(x\) slightly below 1, \(f'(x) > 0\); slightly above 1, \(f'(x) < 0\). So \(x = 1\) is a local maximum.

Step 4: Value
\(f(1) = 1 + (-1)^{2/3} = 1 + 1 = 2\).

Final Answer:
The maximum value is 2. \[ \boxed{\text{(B) }2} \]