First slide

Functions (XII)

Question

The function $f : R \to R$ is defined by $f (x) = \cos^{2} x + \sin^{4} x$ or $x \in R$ Then the range of $f (x)$ is

Moderate

A

$(\frac{3}{4}, 1]$
B

$[\frac{3}{4}, 1)$
C

$[\frac{3}{4}, 1]$
D

$(\frac{3}{4}, 1)$

Solution

$\begin{array}{l} y = f (x) = \cos^{2} x + \sin^{4} x \\ = \cos^{2} x + \sin^{2} x (1 - \cos^{2} x) \\ = \cos^{2} x + \sin^{2} x - \sin^{2} x \cos^{2} x \\ = 1 - \sin^{2} x \cos^{2} x \\ = 1 - \frac{1}{4} \sin^{2} 2 x \\ ∴ \frac{3}{4} \leq f (x) \leq 1 (0 \leq \sin^{2} 2 x \leq 1) \\ ∴ f (x) \in [3 / 4, 1] \end{array}$

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