Applications of Derivatives
Increasing and Decreasing Functions
Grade 12
Question:
<p>Function \(f(x) = 2x^2 - \log|x|, x \neq 0\) monotonically increases in</p>
<p>(a) \(\left(0, \frac{1}{2}\right)\)</p>
<p>(b) \(\left(-\infty, -\frac{1}{2}\right) \cup \left(\frac{1}{2}, \infty\right)\)</p>
<p>(c) \(\left(-\frac{1}{2}, 0\right) \cup \left(\frac{1}{2}, \infty\right)\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Find the derivative and solve $f'(x) > 0$, handling positive and negative $x$ separately.
<p>$f'(x) = 4x - \frac{1}{x}$</p><p>For $x > 0$: $f'(x) > 0 \Rightarrow 4x > \frac{1}{x} \Rightarrow 4x^2 > 1 \Rightarrow x > \frac{1}{2}$</p><p>For $x < 0$: $f'(x) > 0 \Rightarrow 4x - \frac{1}{x} > 0 \Rightarrow 4x^2 > 1 \Rightarrow x < -\frac{1}{2}$</p><p>Thus $f$ increases on $\left(-\infty, -\frac{1}{2}\right) \cup \left(\frac{1}{2}, \infty\right)$</p>
Correct Answer: B