Finding Maxima and Minima using Derivatives

Where is a function at a high or low point? Calculus can help!

A maximum is a high point and a minimum is a low point:

In a smoothly changing function a maximum or minimum is always where the function flattens out  (except for a saddle point).

Where does it flatten out?  Where the slope is zero.

Where is the slope zero?  The Derivative tells us!

Let’s dive right in with an example:

 How Do We Know it is a Maximum (or Minimum)?

We saw it on the graph! But otherwise … derivatives come to the rescue again.

Take the derivative of the slope (the second derivative of the original function):

The Derivative of 14 − 10t is −10

This means the slope is continually getting smaller (−10): travelling from left to right the slope starts out positive (the function rises), goes through zero (the flat point), and then the slope becomes negative (the function falls):


A slope that gets smaller (and goes though 0) means a maximum.

This is called the Second Derivative Test

On the graph above I showed the slope before and after, but in practice we do the test at the point where the slope is zero:

“Second Derivative: less than 0 is a maximum, greater than 0 is a minimum”

 Example: Find the maxima and minima for:

y = 5x3 + 2x2 − 3x

The derivative (slope) is:

y = 15x2 + 4x − 3

Which is quadratic with zeros at:

·         x = −3/5

·         x = +1/3 

Could they be maxima or minima? (Don’t look at the graph yet!)

 The second derivative is y” = 30x + 4

At x = −3/5:

y” = 30(−3/5) + 4 = −14

it is less than 0, so −3/5 is a local maximum

At x = +1/3:

y” = 30(+1/3) + 4 = +14

it is greater than 0, so +1/3 is a local minimum

(Now you can look at the graph.)

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