Applications of Derivatives
Cubic polynomial; roots and inverse trigonometry
Grade Class 12
Question:
Let $f(x)$ be a cubic polynomial such that its local maximum is at $(0,1)$ and local minimum at $(1,-2)$, and $f(-1)<0$, $f(2)>0$. The value of $\sin^{-1}(\cos[\alpha])$ may be equal to (where $\alpha$ is a root of $f(x)=0$, and $[\cdot]$ is the greatest integer function)
$\dfrac{\pi}{2}-1$
$\pi+2$
$2-\pi$
None of these
Step-by-Step Solution
Key Concept: Determine $f(x)$ from the critical point conditions: $f'(x)=3x^2-3x$, so $f(x)=x^3-\frac{3}{2}x^2+1$. Find the integer parts $[\alpha]$ of the roots.
Roots have $[\alpha]=0,1,-1$. $\sin^{-1}(\cos 1)=\frac{\pi}{2}-1$.
Correct Answer: 1