Applications of Derivatives
Local Maxima and Minima
nta_pyq_2024_jan
Grade 12
Question:
The function $f(x)=2x+3(x)^{2/3}$, $x\in\mathbb{R}$, has
exactly one point of local minima and no point of local maxima
exactly one point of local maxima and no point of local minima
exactly one point of local maxima and exactly one point of local minima
exactly two points of local maxima and exactly one point of local minima
Step-by-Step Solution
Key Concept: $f'(x)=2+2x^{-1/3}=2(1+x^{-1/3})=\frac{2(x^{1/3}+1)}{x^{1/3}}$. Critical points: $f'=0$ at $x=-1$, undefined at $x=0$. Sign analysis: $f'>0$ for $x<-1$, $f'<0$ for $-1<x<0$, $f'>0$ for $x>0$. So local max at $x=-1$, local min at $x=0$.
$f'(x)=\frac{2(x^{1/3}+1)}{x^{1/3}}$. Sign: $(+)$ for $x<-1$, $(-)$ for $-1<x<0$, $(+)$ for $x>0$. Local max at $x=-1$, local min at $x=0$. One of each.
Correct Answer: 3