Name: Local Maxima and Minima of a Function
Uploaded: 2019-07-19
Duration: 14 min 17 s
Description: Do you know what is minima or maxima of a function? This video also explains how to find local maxima and minima by differentiation in a detailed manner.

Question

Find the turning points of the following functions and distinguish between them. Also find the local maximum and minimum values of the functions :

$f (x) = x^{3} - 3 x^{2} + 3 x$

Accepted Answer

The given function is,

$f (x) = x^{3} - 3 x^{2} + 3 x . . . . . (1)$

To find turning points (Maxima or minima), we need to use second derivative test.

For this differentiating equation $(1)$ both sides w.r.t. $x$ , we get

$f' (x) = \frac{d}{d x} (x^{3} - 3 x^{2} + 3 x)$

$f' (x) = 3 x^{2} - 6 x + 3 . . . . . . (2)$ $[∵ \frac{d}{d x} x^{n} = n x^{n - 1}]$

On differentiating equation $(2)$ both sides w.r.t $x$ , we get

$f (x) = x^{3} - 3 x^{2} + 3 x$

$f'' (x) = 6 x - 6$ $. . . . . . . . . . . . . . . . (3)$ $[∵ \frac{d}{d x} x^{n} = n x^{n - 1}]$

For turning points(Critical point), Consider $f' (x) = 0$ , then

$3 x^{2} - 6 x + 3 = 0$

$3 (x^{2} - 2 x - 1) = 0$

${(x - 1)}^{2} = 0$

$x = 1$

Thus $x = 1$ is the only critical point.

Put the value of $x$ in $f'' (x)$ ,

$f'' (1) = 6 (1) - 6$

$f'' (1) = 0 . . . . . . . . . . . . . . (4)$

Now calculate $f''' (x)$

$f''' (x) = \frac{d}{d x} (6 x - 6)$

$f''' (x) = 6$

$∵ f'' (1) = 0 a n d f''' (x) = 6 \neq 0$

$∴ x = 1$ is the point of inflection.

So $x = 1$ is neither a point of maxima nor minima.

Find the turning points of the following functions and distinguish between them. Also find the local maximum and minimum values of the functions :

$f (x) = x^{3} - 3 x^{2} + 3 x$

Important Questions on Applications of Derivatives

Learn and Prepare for any exam you want !

Find the turning points of the following functions and distinguish between them. Also find the local maximum and minimum values of the functions : fx=x3-3x2+3x

Important Questions on Applications of Derivatives

Learn and Prepare for any exam you want !

Find the turning points of the following functions and distinguish between them. Also find the local maximum and minimum values of the functions :

$f (x) = x^{3} - 3 x^{2} + 3 x$