Question

A function \[ f(x)=12x^{4/3}-6x^{1/3}, \qquad x\in[-1,1] \] is given. Then, which of the following are TRUE?  

(A) \(f(x)\) has a critical point at \(x=\dfrac{1}{8}\) 
(B) Absolute maximum value of \(f(x)\) is \(18\) 
(C) Absolute maximum value of \(f(x)\) is \(6\) 
(D) Absolute minimum value of \(f(x)\) is \(-\dfrac{9}{4}\)

• A. (A), (C) and (D) only
• B. (A) and (B) only
• C. (A), (B) and (D) only
• D. (C) and (D) only

Hint:

Always include the endpoints of the interval when searching for the absolute maximum and minimum of a function.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To find absolute extrema on a closed interval, we test critical points and endpoints.

Step 2: Mathematical Calculation:
\( f'(x) = 12(4/3)x^{1/3} - 6(1/3)x^{-2/3} = 16x^{1/3} - 2x^{-2/3} \).
Set \( f'(x) = 0 \Rightarrow 16x^{1/3} = \frac{2}{x^{2/3}} \Rightarrow 16x = 2 \Rightarrow x = 1/8 \).
Evaluate: \( f(1/8) = 12(1/16) - 6(1/2) = 0.75 - 3 = -2.25 \) (which is \( -9/4 \)).
Evaluate endpoints: \( f(-1) = 12(1) - 6(-1) = 18 \), \( f(1) = 12 - 6 = 6 \).
Max is 18, Min is -2.25. Statements (A), (B), (D) are true.

Step 3: Final Answer:
The correct combination is (c).