Question

The interval in which $f(x) = 2x^{2} - \log x$ increases is

• A. $(-1/2, 0)$
• B. $(0, 1/2)$
• C. $(-1/2, 1/2)$
• D. $(1/2, \infty)$

Hint:

Always check the domain first. For logarithmic functions, the argument must be strictly positive ($x > 0$).

Answer 1

The Correct Option is D

Step 1: Concept
A function increases where its first derivative $f'(x)$ is greater than zero.

Step 2: Meaning
Calculate the derivative: $f'(x) = 4x - \frac{1}{x}$. Note that for $\log x$, $x$ must be greater than 0.

Step 3: Analysis
Set $f'(x) > 0 \implies 4x - \frac{1}{x} > 0 \implies \frac{4x^{2}-1}{x} > 0$. Since $x > 0$, we solve $4x^{2} - 1 > 0 \implies x^{2} > 1/4$.

Step 4: Conclusion
$x^{2} > 1/4$ means $x > 1/2$ or $x < -1/2$. Given the domain $x > 0$, the function increases in the interval $(1/2, \infty)$. Final Answer: (D)