yourself. Once you have the equation, you find its derivative and set it equal to zero. The values you get
are called critical values. That is, if f ́(c) = 0 or f ́(c) does not exist, then c is a critical value. Then, test
these values to determine whether each value is a maximum or a minimum. The simplest way to do this is
with the second derivative test.
If a function has a critical value at x = c, then that value is a relative maximum if f ́ ́(c) < 0 and it is a
relative minimum if f ́ ́(c) > 0.If the second derivative is also zero at x = c, then the point is neither a maximum nor a minimum but a
point of inflection. More about that later.
It’s time to do some examples.
Example 1: Find the minimum value on the curve y = ax^2 , if a > 0.
Take the derivative and set it equal to zero.
= 2ax = 0The first derivative is equal to zero at x = 0. By plugging 0 back into the original equation, we can solve
for the y-coordinate of the minimum (the y-coordinate is also 0, so the point is at the origin).
In order to determine if this is a maximum or a minimum, take the second derivative.
= 2aBecause a is positive, the second derivative is positive and the critical point we obtained from the first
derivative is a minimum point. Had a been negative, the second derivative would have been negative and
a maximum would have occurred at the critical point.
Example 2: A manufacturing company has determined that the total cost of producing an item can be
determined from the equation C = 8x^2 − 176x + 1,800, where x is the number of units that the company
makes. How many units should the company manufacture in order to minimize the cost?
Once again, take the derivative of the cost equation and set it equal to zero.
= 16x − 176 = 0x = 11