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 points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be:

$f (x) = \sin x - \cos x, 0 < x < 2 π$

Accepted Answer

Given : $f (x) = \sin x - \cos x, 0 < x < 2 π$

$\Rightarrow f' (x) = cosx + sinx$

For a local maximum or a local minimum, we must have

$f' (x) = 0$

$\Rightarrow cosx + sinx = 0$

$\Rightarrow cosx = - sinx$

$\Rightarrow tanx = - 1$

$\Rightarrow \tan x = \tan (- \frac{π}{4})$

We know that the general solution of $\tan x = \tan y$ is $x = n π + y,$ where $n \in Z$ .

Thus, we have $x = n π - \frac{π}{4}, n \in Z$

Now for $0 < x < 2 π$ , we have

$x = \frac{3 π}{4}$ and $\frac{7 π}{4}$

So, the critical points of the given function are $\frac{3 π}{4}, \frac{7 π}{4}$

Now, for $0 < x < \frac{3 π}{4}, f^{'} (x) > 0$ , and for $\frac{3 π}{4} < x < \frac{7 π}{4}, f^{'} (x) < 0,$ and for $x > \frac{7 π}{4}, f^{'} (x) > 0$

Since $f' (x)$ changes from positive to negative when $x$ increases through $\frac{3 π}{4},$

$∴ x = \frac{3 π}{4}$ is the point of local maxima.

The local maximum value of $f (x)$ at $x = \frac{3 π}{4}$ is given by

$f (\frac{3 π}{4}) = \sin (\frac{3 π}{4}) - \cos (\frac{3 π}{4}) = \sqrt{2}$

Since $f' (x)$ changes from negative to positive when $x$ increases through $\frac{7 π}{4}, x = \frac{7 π}{4}$ is the point of local minima.

The local minimum value of $f (x)$ at $x = \frac{7 π}{4}$ is given by

$\sin (\frac{7 π}{4}) - \cos (\frac{7 π}{4}) = \sin (2 π - \frac{π}{4}) - \cos (2 π - \frac{π}{4}) = - \sin \frac{π}{4} - \cos \frac{π}{4} = - \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} = - \sqrt{2}$

Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be:

$f (x) = \sin x - \cos x, 0 < x < 2 π$

Important Questions on Maxima and Minima

Learn and Prepare for any exam you want !

Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be: f(x)=sinx-cosx,0<x<2π

Important Questions on Maxima and Minima

Learn and Prepare for any exam you want !

Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be:

$f (x) = \sin x - \cos x, 0 < x < 2 π$