Question

Find the maximum and minimum value of the function:
$f\left(x\right)=2x^3-21x^2+36x-20$.

Answer 1

$f\left(x\right)=2x^3-21x^2+36x-20$

Any cubic polynomial has 2 stationary points which are nothing but the points of local maxima and local minima. These are at $f^{\prime }\left(x\right)=0$

Hence, on differentiating, we get

$f^{\prime }\left(x\right)=6x^2-42x+36=0$

$\therefore x^2-7x+6=0$

$\therefore x^2-6x-x+6=0$

$\therefore (x-1)(x-6)=0$

$\therefore x=1\text{or}x=6$

Now,

$f\left(1\right)=2-21+36-20=-3$

$f\left(6\right)=432-756+216-20=-128$

Hence, local maxima and local minima of $f\left(x\right)$ are at $x=1$ and $x=6$ with value $-3$ and $-128$ respectively.