Functions
Functions
Allen Star Batch
Grade 12
Question:
Let $f(x) = \sin^3 x - \sin x \cos x + \cos^3 x$, then range of $f(x)$ is:
$\left[0, \frac{9}{8}\right]$
$\left[-\frac{9}{8}, 0\right]$
$\left(0, \frac{9}{8}\right)$
$\left(-\frac{9}{8}, 0\right)$
Step-by-Step Solution
Key Concept: Express f(x) = sin³x - sin x cos x + cos³x using the identity (sin x + cos x)³ and double angle formulas to reduce it to a quadratic function g(t) = 1 - ½t - ½t² where t = sin 2x ∈ [-1,1], then find the range by completing the square.
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