First slide

Extreme values and Periodicity of Trigonometric functions

Question

If $- \frac{π}{2} < θ < \frac{π}{2}$ , then the minimum value of $\cos^{3} θ + \sec^{3} θ$ is

Moderate

A

$1$
B

$2$
C

$0$
D

none of these.

Solution

We know that
A.M. $\geq$ G.M.
$∴ \frac{\cos^{3} θ + \sec^{3} θ}{2} \geq \sqrt{\cos^{3} θ \times \sec^{3} θ}$
$\Rightarrow \cos^{3} θ + \sec^{3} θ \geq 2$ .
Hence, the minimum value of $\cos^{3} θ + \sec^{3} θ$ is $2$ .

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