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

Determine the points of maxima and minima of the function

f (x) = \frac{1}{8} \ln x - b x + x^{2}, x > 0,

where

b \geq 0

is constant.

Accepted Answer

Given:

$f (x) = \frac{1}{8} \log x - b x + x^{2}$

Differentiating on both side with respect to $x$ , we get

$b \geq 0$

$\Rightarrow f' (x) = \frac{1 - 8 b x + 16 x^{2}}{8 x}$

For the extreme points

$f' (x) = 0$

$\Rightarrow \frac{1 - 8 b x + 16 x^{2}}{8 x} = 0$

$\Rightarrow 16 x^{2} - 8 b x + 1 = 0$

$\Rightarrow x = \frac{8 b \pm \sqrt{{(- 8 b)}^{2} - 4 \times 16 \times 1}}{2 \times 16}$

$\Rightarrow x = \frac{b \pm \sqrt{b^{2} - 1}}{4}$

Let $α = \frac{b - \sqrt{b^{2} - 1}}{4}$ and $β = \frac{b + \sqrt{b^{2} - 1}}{4}$

For the real roots discriminant $\geq 0$

$\Rightarrow b^{2} - 1 \geq 0$

$\Rightarrow b \geq 1$ or $b \leq - 1$ .

Draw a number line for $f' (x)$ .

According to number line when point moves from positive to negative that is point of maxima, and if point moves from negative to positive then it is point of minima

So,

$f (x)$ is maximum at $x = \frac{b - \sqrt{b^{2} - 1}}{4}$

$f (x)$ is minimum at $x = \frac{b + \sqrt{b^{2} - 1}}{4}$

Let $b = 1$ , then

$f' (x) = \frac{16 x^{2} - 8 (1) x + 1}{8 x}$

$\Rightarrow f' (x) = \frac{{(4 x - 1)}^{2}}{8 x}$

Given $x > 0$ , so

$f' (x) = \frac{{(4 x - 1)}^{2}}{8 x} > 0$

So,

$f (x)$ neither minimum nor maximum for $b = 1$

Determine the points of maxima and minima of the function $f (x) = \frac{1}{8} \ln x - b x + x^{2}, x > 0,$ where $b \geq 0$ is constant.

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Determine the points of maxima and minima of the function f(x)=18lnx-bx+x2,x>0, where b≥0 is constant.

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Determine the points of maxima and minima of the function $f (x) = \frac{1}{8} \ln x - b x + x^{2}, x > 0,$ where $b \geq 0$ is constant.