Find the minimum value of 2 + 1 + 2 0 < < 2.

By AM-GM, 2 + 1 + 2 3 [3]2 1 2 = 3 2.Equality occurs when = 4, so the minimum value is 3 2.

jmc

algebra intermediate

Problem

Find the minimum value of

2 cos θ + \frac{1}{sin θ} + 2 tan θ

for

0 < θ < \frac{π}{2} .

Solution

By AM-GM,

2 cos θ + \frac{1}{sin θ} + 2 tan θ \geq 3 3 2 cos θ \cdot \frac{1}{sin θ} \cdot 2 tan θ = 32 .

Equality occurs when

θ = \frac{π}{4},

so the minimum value is

32 .

Final answer

3 \sqrt{2}