First slide

Theory of expressions

Question

If $α \in (0, π / 2),$ then the expression $\sqrt{x^{2} + x} + \frac{\tan^{2} α}{\sqrt{x^{2} + x}}$ is always greater than or equal to

Moderate

A

$2 \tan α$
B

2
C

1
D

$s e c^{2} α$

Solution

As A.M $\geq$ G.M, we get
$\begin{matrix} \sqrt{x^{2} + x} + \frac{\tan^{2} α}{\sqrt{x^{2} + x}} \geq 2 {[\sqrt{x^{2} + x} \frac{\tan^{2} α}{\sqrt{x^{2} + x}}]}^{1 / 2} \\ = 2 \tan α \end{matrix}$

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