Question

In equation $3^{si{n}^2\theta }+3^{co{s}^2\theta }\ge 2\sqrt{3}$ is true

• A. for all real values of $\theta$
• B. some real values of $\theta$
• C. for imaginary values of $\theta$
• D. for no value of $\theta$

Answer 1

The correct option is A for all real values of $\theta$

$AM\ge GM\Rightarrow \frac{3^{si{n}^2\theta }+3^{co{s}^2\theta }}{2}\ge \sqrt{3^{si{n}^2\theta }{.3}^{co{s}^2\theta }}=\sqrt{3^{(si{n}^2\theta +co{s}^2\theta )}}=\sqrt{3}$

$\frac{3^{si{n}^2\theta }+3^{co{s}^2\theta }}{2}\ge \sqrt{3}\Rightarrow 3^{si{n}^2\theta }+3^{co{s}^2\theta }\ge 2\sqrt{3}$

Therefore, It is True for all values of $\theta$