## ExtremE ValuEs

Probably the most important application of derivatives is to find the extreme values of a function. Using derivatives, one can readily convert the problem of finding the extreme value into solving an equation. Then other methods, including using calculus again, can be employed.

So. Think about the derivative as the slope of the tangent line at the given point. What if the slope is 0?

This could only mean one thing: the tangent line is flat at that point. It must be a peak or a trough.

And peaks and troughs are what we are looking for if we want to find extreme values of functions. Is that right?

Now there is just two things left to do: examine whether this is a peek or trough, then compare it to the other peaks or troughs found also by solving the equation: derivative=0.

However, consider the following scenario: what if we have found a peak, and it is the only peak we found. Is it sure to be the maximum of the function? NO.

Consider the following graph:

So. Think about the derivative as the slope of the tangent line at the given point. What if the slope is 0?

This could only mean one thing: the tangent line is flat at that point. It must be a peak or a trough.

And peaks and troughs are what we are looking for if we want to find extreme values of functions. Is that right?

Now there is just two things left to do: examine whether this is a peek or trough, then compare it to the other peaks or troughs found also by solving the equation: derivative=0.

However, consider the following scenario: what if we have found a peak, and it is the only peak we found. Is it sure to be the maximum of the function? NO.

Consider the following graph:

The two black lines at either side means it is cut off from a bigger curve. Now suppose we would like to find the maximum value in this section. Obviously, there is only one peak, and that is in the middle. However, its value is no match compared to the values on either side of the selected curve. why?

This is because on either side, the function could be increasing rapidly, but was cut off. In the middle, the function could be in a transition from a low growth into a low decay.

So, this brings something more into our equation: the value of the function on both endpoints of the curve we are studying.

Another thing to consider is gaps and isolated points in the curve. suppose we have a parabola: y=x^2. (when x=0, y=-1) Everything is nice. We have the derivative 2x, we solve it and find the minimum to occur at x=0, and y=0.

Look at the parentheses next to y=x^2. You will find that y=-1 when x=0. since -1<0, we safely conclude out smallest value is -1. However, this is a lucky scenario. What if y=1 at x=0? Then the minimum no longer occurs at x=0. it occurs on either side of it. (it is noncontinuous at x=0.)

Below is a step by step approach to attack extreme value problems:

1. Calculate derivative.

(usually the easiest in real-life)

2. Solve "derivative=0".

(usually the toughest in real-life)

3. Compare the values obtained by evaluating the function at the x's that make the derivative 0 with the values of x's that are on the boundary lines, or make the function noncontinuous.

4. Conclude your results.

This is because on either side, the function could be increasing rapidly, but was cut off. In the middle, the function could be in a transition from a low growth into a low decay.

So, this brings something more into our equation: the value of the function on both endpoints of the curve we are studying.

Another thing to consider is gaps and isolated points in the curve. suppose we have a parabola: y=x^2. (when x=0, y=-1) Everything is nice. We have the derivative 2x, we solve it and find the minimum to occur at x=0, and y=0.

**WRONG!**Look at the parentheses next to y=x^2. You will find that y=-1 when x=0. since -1<0, we safely conclude out smallest value is -1. However, this is a lucky scenario. What if y=1 at x=0? Then the minimum no longer occurs at x=0. it occurs on either side of it. (it is noncontinuous at x=0.)

Below is a step by step approach to attack extreme value problems:

1. Calculate derivative.

(usually the easiest in real-life)

2. Solve "derivative=0".

(usually the toughest in real-life)

3. Compare the values obtained by evaluating the function at the x's that make the derivative 0 with the values of x's that are on the boundary lines, or make the function noncontinuous.

4. Conclude your results.

*And now you can brag that you have finished studying the most important application of derivatives!*