Question

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

Answer 1

We have,

$f\left(x\right)=x^5-5x^4+5x^3-1$

On differentiating and we get,

$f^{\prime }\left(x\right)=5x^4-20x^3+15x^2$

For maxima and minima

$f^{\prime }\left(x\right)=0$

$5x^4-20x^3+15x^2=0$

$\Rightarrow 5x^2(x^2-4x+3)=0$

$\Rightarrow x^2-4x+3=0$

So,

$x^2-(3+1)x+3=0$

$\Rightarrow x^2-3x-1x+3=0$

$\Rightarrow x(x-3)-1(x-3)=0$

$\Rightarrow (x-3)(x-1)=0$

If,

$x-3=0$

$x=3$

If,

$x-1=0$

$x=1$

Again differentiating and we get,

$f^{\prime \prime }\left(x\right)=20x^3-60x^2+30x$

Put $x=3$

$f^{\prime \prime }\left(3\right)=20\times {\left(3\right)}^3-60\times {\left(3\right)}^2+30\times 3$

$f^{\prime \prime }\left(3\right)=540-540+90$

$f^{\prime \prime }\left(3\right)=90\left(\text{minima}\right)$

If put $x=1$ and we get,

$f^{\prime \prime }\left(1\right)=20{\left(1\right)}^3-60{\left(1\right)}^2+30\left(1\right)$

$f^{\prime \prime }\left(1\right)=-10\left(\text{maxima}\right)$

Hence, this is the answer.