Applications of Derivatives
Monotonicity and Zero Counting
nta_pyq_2024_apr
Grade 12
Question:
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 $\left[0,\dfrac{\pi}{2}\right]$ and increasing in $\left[\dfrac{\pi}{2},\pi\right]$.
Both (S1) and (S2) are correct.
Both (S1) and (S2) are incorrect.
Only (S2) is correct.
Only (S1) is correct.
Step-by-Step Solution
Key Concept: $f'(x)=-\sin x-1<0$ always: $f$ strictly decreasing. S2 false. One zero by IVT: S1 true.
S1 ✓, S2 ✗.
Correct Answer: 4