It’s Professor Dave, let’s find maxima and minima.
By now we have a solid concept of differentiation.
What it is, and how to do it, with a variety of functions.
So let’s continue to learn 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, with local maxima and minima, then 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.
There are countless applications for this, and we will go over many of them a little
later when we look at optimization problems.
For now, let’s not worry about application, let’s just get some practice in finding
these values.
Of course, some functions have no maxima or minima.
Take x cubed for example.
This function is constantly increasing as we move left to right, from negative infinity
to positive infinity, so there are no maxima or minima to speak of.
Some functions have absolute maxima and minima.
This would be something like sine of x.
This function will never be greater than one, which is its value at any input that is half
pi plus or minus any multiple of two pi.
Likewise, it will never be less than negative one, which is its value at any of these inputs.
So these are absolute maxima and minima.
If we zoom in on a specific portion, we can identify a local, or relative maximum or minimum,
but this will repeat in either direction.
Some functions instead will have a finite number of local maxima or minima.
This 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.
Let’s look at the function x cubed minus three x squared plus one.
There is one local maximum and one local minimum.
Where do these occur?
Well as we said, let’s take the derivative.
That will be three x squared minus six x.
We can factor out a three x from both terms to get three x times (x minus two).
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.
If x is zero, f prime is zero.
If x is two, f prime is zero.
So the local maximum occurs at x equals zero, and the local minimum occurs at x equals two.
Of course that one was pretty simple, so let’s try one more for practice.
What about x over the quantity x squared plus one?
Again, we must take the derivative, and for that we need the quotient rule.
Remember it’s bottom times the derivative of the top, minus top times the derivative
of the bottom, over bottom squared.
Here that gives us this, and we can simplify to get negative x squared plus one on top.
Now remember, to find the zeros of a rational function like this one, we just need to find
the zeros of the numerator, since zero over anything is zero, so let’s set negative
x squared plus one equal to zero.
We rearrange to get x squared equals one, take the square root of both sides, and x
must be equal to plus or minus one.
That means that x equals negative one and one will give local maxima or minima on the
function, and in fact, the function looks like this.
So this is one great application of differentiation.
In algebra we learned how to get a rough sketch of a higher-degree polynomial, by finding
the end behavior and its zeros, but it was always a pretty rough sketch, as we never
knew precisely how far above or below the x axis to go after each pass.
But now we can even find local maxima and minima, and then evaluate the function at
these input values, which gives additional important points on the curve, so this new
trick really allows us to draw near flawless graphs for even these tricky higher-degree functions.
Speaking of graphing functions, let’s revisit this in the context of calculus, but first,
let’s check comprehension.