Question

36. For the function f(x) = (cosx)-x+1, x ∈ R, between the following two statements:

Statement-1 : f(x) = 0 for only one value of x in [0, π].

Statement-2 : f(x) is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$.

• A. Both Statemant-1 and Statement-2 are correct
• B. Only Statement-2 is correct
• C. Both Statemant-1 and Statement-2 are incorrect
• D. Only Statement-1 is correct

Answer 1

Answer: (D)

Only Statement-1 is correct

Answer 2

Statement-1 is correct because $f(0) = 2$ and $f(\pi) = -\pi$. Since $f(x)$ is continuous and changes sign, there is at least one root. The derivative $f'(x) = -\sin x - 1$ is always negative for $x \in [0, \pi]$, so $f(x)$ is strictly decreasing and has exactly one root. Statement-2 is incorrect because $f'(x) = -\sin x - 1$ is always negative for $x \in [0, \pi]$, meaning $f(x)$ is decreasing on both intervals, not increasing on the second.