Differentiation:
By now we have a solid concept of differentiation and how to do it with a variety of functions.
So Let's continue to see about some applications for differentiation how is it useful and what can it tell us about a function well if a function looks like this graphically with local maximum and minimum value of a function,
A derivative can be very useful in figuring out where those are exactly.
Let's recall that if a derivative can be interpreted as the slope of a tangent line and the tangent line at any maximum or minimum must be perfectly horizontal then the derivative of a function at any of these points must equal zero.
So to find them all we need to do is take the derivative and find the zeros of the derivative.
Let's just get some practice in finding these values
No Maxima and Minima of Function:
Some functions have no maxima or minima
Consider we have a function y=x3
The maximum and minimum graph of a function is given as:
This function is constantly increasing as we move left to right from negative infinity to positive infinity.
So there are no maxima or minima of functions like this.
Absolute Maxima and Minima of Function:
Some functions have absolute maxima and minima values. Let's just see in an example
Consider we have a function y= sin x
The maximum and minimum graph of a function is given as:
This function will never be greater than 1,which is its value at any domain inputs.
Likewise it will never be less than -1 which is its value at any of domain inputs.
So these are absolute maxima and minima, if we zoom in on a specific portion we can identify a local or relative maxima or minima but this will repeat in either direction.
Exact Maxima and Minima of Function:
Some functions instead of having a finite number of local Maxima or Minima, there is just the highest or lowest point on the curve in a particular section which we can identify as any point where the function changes direction.
Consider we have a function
$$ f(x)= x^3-3x^2+1 $$
The maximum and minimum graph of a functin is given as:
There is one local maximum and one local minimum where these occur?
For that let's take the derivative that will be:
$$ f'(x) = 3x^2 - 6x $$ $$ f'(x) = 3x(x-2) $$
Now we want to find the input values for which this derivative equals zero because those will be the points on the function with horizontal tangent lines well this is just basic algebra.
i.e
$$ f'(x)=0 $$ $$ 3x(x-2)=0 $$ $$ 3x=0 \;\;\;\;,\;\;\;\; x-2=0 $$ $$ x=0 \;\;\;\;,\;\;\;\; x=2 $$ Now, if x=0 and x=2 then also f'(x)=0 . $$ \implies \;\;\;\; f'(0)=0 \;\;\;\;,\;\;\;\; f'(2)=0 $$
So the local maximum occurs at x=0 and the local minimum occurs at x=2.
These are the basic concepts of finding abosulute and relative maxima and minima using a differentiation method which is quite simple.