Question

The function $f\left(x\right)=4x^5-25x^4+40x^3-10$ has

• A. two local maximum and two local minimum
• B. one local maximum and one local minimum
• C. one local maximum and two local minimum
• D. two local maximum and one local minimum

Answer 1

The correct option is B one local maximum and one local minimum

$f\left(x\right)=4x^5-25x^4+40x^3-10$
$\Rightarrow f^{\prime }\left(x\right)=20x^4-100x^3+120x^2$
From $\Rightarrow f^{\prime }\left(x\right)=0$
$\Rightarrow 20x^2(x^2-5x+6)=0$
$\Rightarrow 20x^2(x-3)(x-2)=0$
$\Rightarrow x=0,2,3$

From the above sign change, we get that there is no change with respect to $x=0$ So, at $x=0,f\left(x\right)$ has neither local maxima nor local minima and at $x=2,3\Rightarrow$ one local maximum and one local minimum respectively.