Which of the following statements are true? (A) The function f(x)=(x^…

The correct answer is Option (3) → (A) and (D) only

$\text{(A)}\;f(x)=\frac{x^4}{4}-\frac{4}{3}x^3+\frac{x^2}{2}+6x.$

$f'(x)=x^3-4x^2+x+6.$

$x^3-4x^2+x+6=(x-2)(x^2-2x-3)=(x-2)(x-3)(x+1).$

$\text{Critical points: }x=-1,2,3.$

$\Rightarrow \text{3 critical points.}$

$\text{(A) True.}$

$\text{(B)}\;f(x)=|x|+3.$

$\text{Minimum occurs at }x=0,\;f(0)=3.$

$\Rightarrow \text{Statement is false.}$

$\text{(C)}\;\text{Local maximum need not be absolute maximum.}$

$\Rightarrow \text{False.}$

$\text{(D)}\;f(x)=x^2.$

$f'(x)=2x=0 \Rightarrow x=0.$

$f''(x)=2>0 \Rightarrow \text{minimum at }x=0.$

$\Rightarrow \text{True.}$

$\text{Correct statements: (A) and (D).}$