Question

Find the points of local maxima, local minima and the points of inflection of the function f(x)= $x^5-5x^4+5x^3-1.$ Also find the corresponding local maximum and local minimum values.

Answer 1

Given that, $f\left(x\right)=x^5-5x^4+5x^3-1$
On differentiating w.r.t. x, we get
$f^{\prime }\left(x\right)=5x^4-20x^3+15x^2$
For maxima or minima, $f^{\prime }\left(x\right)=0\Rightarrow 5x^4-20x^3+15x=0\Rightarrow 5x^2(x^2-4x+3)=0\Rightarrow 5x^2(x^2-3x-x+3)=0\Rightarrow 5x^2\left[x\right(x-3)-1(x-3\left)\right]=0\Rightarrow 5x^2\left[\right(x-1\left)\right(x-3\left)\right]=0\therefore x=0,1,3$
Sign scheme for $\frac{dy}{dx}=5x^2(x-1)(x-3)$

So, y has maximum value at x = 1 and minimum value at x = 3. At x = 0, y has neither maximum nor minimum value.
$\therefore$ Maximum value of $y=1-5+5-1=0$
and minimum value $=(3{)}^5-5(3{)}^4+5(3{)}^3-1=243-81\times 5-27\times 5-1=-298$