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The function f (x) = tanx - x
A. always increases
B. always decreases
C. never increases
D. sometimes increases and sometimes decreases.

Answers (1)

Given f (x) = tanx - x

Apply first derivative and get

\mathrm{f}^{\prime}(\mathrm{x})=\frac{\mathrm{d}(\tan \mathrm{x}-\mathrm{x})}{\mathrm{dx}}

Apply sum rule and get
\Rightarrow f^{\prime}(x)=\frac{d(\tan x)}{d x}-\frac{d(x)}{d x}

Apply derivative,
\Rightarrow f^{\prime}(x)=\sec ^{2} x-1
Square of every number is always positive,
So f^{\prime}(x)>0 \forall x \in R
So f (x) = tanx - x always increases.
So the correct answer is option A

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