For the function $f(x)=(\cos x)-x+1, x \in \mathbb{R}$, between the following two statements (S1) $f(x)=0$…
For the function $f(x)=(\cos x)-x+1, x \in \mathbb{R}$, between the following two statements
(S1) $f(x)=0$ for only one value of $x$ in $[0, \pi]$.
(S2) $f(x)$ is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$.
Both (S1) and (S2) are correct.
Both (S1) and (S2) are incorrect.
Only (S2) is correct.
Only (S1) is correct.
Solution
$\begin{aligned} & f^{\prime}(x)=\cos x-x+1 \\ & f^{\prime}(x)=-\sin x-1\end{aligned}$
$\mathrm{f}$ is decreasing $\forall \mathrm{x} \in \mathrm{R}$
$\begin{aligned} & \mathrm{f}(\mathrm{x})=0 \\ & \mathrm{f}(0)=2, \mathrm{f}(\pi)=-\pi\end{aligned}$
$\mathrm{f}$ is strictly decreasing in $[0, \pi]$ and $\mathrm{f}(0) . \mathrm{f}(\pi) < 0$
$\Rightarrow$ only one solution of $\mathrm{f}(\mathrm{x})=0$
$\mathrm{S} 1$ is correct and $\mathrm{S} 2$ is incorrect.