Question

Let $f$ be the function $f\left(x\right)=\cos x-(1-\frac{x^2}{2})$, then find the interval in which $f\left(x\right)$ is strictly increasing.

• A. $(-\infty ,\infty )$
• B. $(-2,\infty )$
• C. $[0,\infty )$
• D. $(0,\infty )$

Answer 1

The correct option is D $(0,\infty )$

$f\left(x\right)=\cos x-(1-\frac{x^2}{2})$

$\Rightarrow f^{\prime }\left(x\right)=-\sin x+x$

Given that this function is strictly increasing. Hence,

$\Rightarrow f^{\prime }\left(x\right)>0$

$\Rightarrow -\sin x+x>0$

$\Rightarrow x>sinx$

Therefore, $x$ belongs to $(0,\infty )$