Sets, Relations & Functions
Functions
star_batch_jee_advanced_2025
Grade 11
Question:
Let $f(x) = \sin^3 x - \sin x \cos x + \cos^3 x$, then range of $f(x)$ is:
[0, 9/8]
[-9/8, 0]
(0, 9/8)
(-9/8, 0)
Step-by-Step Solution
Key Concept: Completing the square after substitution reveals the extrema of the function.
Simplify $f(x) = 1 - \frac{1}{2}\sin 2x - \frac{1}{2}\sin^2 2x = 1 - \frac{1}{2}t - \frac{1}{2}t^2$ where $t = \sin 2x \in [-1,1]$. This equals $\frac{9}{8} - \frac{1}{2}(t+\frac{1}{2})^2$.
Correct Answer: 1