Application of Derivatives 2 Question 30
30. Show that $2 \sin x+2 \tan x \geq 3 x$, where $0 \leq x<\pi / 2$.
(1990, 4M)
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Answer:
Correct Answer: 30. (a, c)
Solution:
- Let $y=f(x)=2 \sin x+2 \tan x-3 x$
$\Rightarrow \quad f^{\prime}(x)=2 \cos x+2 \sec ^{2} x-3$
For $\quad 0 \leq x<\pi / 2, f^{\prime}(x)>0$
Thus, $f(x)$ is increasing.
$$ \begin{array}{ll} \text { When } & x \geq 0, f(x) \geq f(0) \\ \Rightarrow & 2 \sin x+2 \tan x-3 x \geq 0+0-0 \\ \Rightarrow & 2 \sin x+2 \tan x \geq 3 x \end{array} $$
