Question

The function

f
(
x
)
=
2
x
3
−
27
x
2
+
108
x
−
9
has two critical numbers.

The smaller one is x =

and the larger one is x =

Answer 1

• The smaller one is x = 3
• and the larger one is x = 6

Explanation:

A "critical number" is a function argument where the function has zero or undefined slope. A polynomial function never has undefined slope, so we're looking for the x-values where f'(x) = 0.

A graphing calculator easily shows the points that have zero slope. They are found at x = 3 and x = 6.

_____

Setting the derivative to zero is another way to find the points with zero slope:

f'(x) = 6x^2 -54x +108 = 6(x^2 -9x +18) = 6(x-3)(x-6)

The derivative will be zero where the factors are zero, at x=3 and x=6.

The critical numbers are x=3 and x=6.