Definite Integration
Monotonicity of g(θ) from integral definition
MJAT_TS8_P2
Grade 12
Question:
A function $g(\theta)=\displaystyle\int_0^{\sin^2\theta}f(x)\,dx+\int_0^{\cos^2\theta}f(x)\,dx$ is defined on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ where $f(x)$ is increasing. Then $g(\theta)$ is increasing on:
A) $\left(-\dfrac{\pi}{2},0\right)$
B) $\left(\dfrac{\pi}{4},\dfrac{\pi}{2}\right)$
C) $\left(0,\dfrac{\pi}{4}\right)$
D) $\left(-\dfrac{\pi}{4},0\right)$
Step-by-Step Solution
Key Concept: $g'(\theta)=(f(\sin^2\theta)-f(\cos^2\theta))\sin 2\theta$. For $g'>0$: either both factors positive or both negative.
Answer: **D**.
Correct Answer: D