Let $f(x) = 4x^3- 18x^2 +27x - 5,x∈R$. Then which of the following statements are TRUE?
(A) $f''(x) = 24x - 36$
(B) $f$ has local maxima at $x =\frac{3}{2}$ but no minima
(C) $f$ has neither maxima nor minima
(D) $f$ has both maxima and minima
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
$f(x)=4x^{3}-18x^{2}+27x-5$
$f'(x)=12x^{2}-36x+27=12\left(x-\frac{3}{2}\right)^{2}\ge 0$
$f''(x)=24x-36$
Critical point: $f'(x)=0 \Rightarrow x=\frac{3}{2}$ and $f''\!\left(\frac{3}{2}\right)=0$.
Since $f'(x)\ge 0$ for all $x$ and vanishes only at $x=\frac{3}{2}$, $f$ is strictly increasing with a stationary inflection; no local extrema.
True statements: (A) and (C)