Question

The function $f(x) = x^4 - 2x^2$ is strictly increasing on

• A. $(-2,0)$ and $[1,\infty)$
• B. $[-1,0]$ and $[2,\infty)$
• C. $[-1,0]$ and $[1,\infty)$
• D. $(-2,0]$ and $[0,\infty)$
• E. $[-2,0]$ and $(1,\infty)$

Hint:

Factor derivatives completely to make sign analysis easy.

Answer 1

The Correct Option is C

Concept: A function is strictly increasing where: \[ f'(x) > 0 \]

Step 1: Find derivative
\[ f(x) = x^4 - 2x^2 \] \[ f'(x) = 4x^3 - 4x = 4x(x^2 - 1) \] \[ = 4x(x-1)(x+1) \]

Step 2: Find critical points
\[ x = -1,\ 0,\ 1 \]

Step 3: Sign analysis
\[ \begin{array}{c|c} \text{Interval} & f'(x) \hline (-\infty,-1) & - (-1,0) & + (0,1) & - (1,\infty) & + \end{array} \]

Step 4: Conclusion
Function is increasing where $f'(x) > 0$: \[ (-1,0) \text{ and } (1,\infty) \] Including endpoints: \[ [-1,0] \text{ and } [1,\infty) \] Final Conclusion:
Option (C)