Applications of Derivatives
Rolle's Theorem / Counting Zeros
nta_pyq_2024_apr
Grade 12
Question:
Let $f:\mathbb{R}\to\mathbb{R}$ be a thrice differentiable function such that $f(0)=0,\ f(1)=1,\ f(2)=-1,\ f(3)=2$ and $f(4)=-2$. Then, the minimum number of zeros of $(3f'f''+ff''')(x)$ is _______.
Step-by-Step Solution
Key Concept: Observe $3f'f''+ff'''=(f\cdot f')''$. $f$ has at least 4 zeros (sign changes). By Rolle's: $f'$ min 3, $ff'$ min 7, $(ff')'$ min 6, $(ff')''$ min 5.
Min zeros of $(ff')''=5$.
Correct Answer: 5