Question

Find maxima and minima of a function problems

Answer 1

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:

quadratic graph

Example: A ball is thrown in the air. Its height at any time t is given by:

h = 3 + 14t − 5t2

What is its maximum height?

Using derivatives we can find the slope of that function:

d/dth = 0 + 14 − 5(2t)

= 14 − 10t

(See below this example for how we found that derivative.)

quadratic graph

Now find when the slope is zero:

14 − 10t = 0

10t = 14

t = 14 / 10 = 1.4

The slope is zero at t = 1.4 seconds

And the height at that time is:

h = 3 + 14×1.4 − 5×1.42

h = 3 + 19.6 − 9.8 = 12.8

And so:

The maximum height is 12.8 m (at t = 1.4 s)