Calculus I: Derivative
First Derivative Test

1. The First Derivative Test

Suppose f is continuous at a critical point xo.

• If f'(x) > 0 on an open interval extending left from xo and f'(x) < 0 on an open interval extending right from xo, then f has a relative maximum at xo.

• If f'(x) < 0 on an open interval extending left from xo and f'(x) > 0 on an open interval extending right from xo, then f has a relative minimum at xo.

• If f'(x) has the same sign on both an open interval extending left from xo and an open interval extending right from x0, then f does not have a relative extremum at xo.

• In summary, relative extrema occur where f(x) changes sign.

2. Example

Let's consider, on the interval [- 2, 3], the following function:

f(x) = 3 x⁴ - 4 x³- 12 x² + 3

This function is differentiable everywhere on the interval [- 2, 3].

Taking the derivative, we obtain:

f'(x) = 12 x³ - 12 x²- 24 x = 12 x ( x² - x - 2) = 12 x (x - 2)(x + 1)

f(x) = 0 for x = - 1, 0, and + 2.

These are the three critical points of f on [- 2, 3].

x	- ∞	- 1	0	2	+ ∞
x + 1	-	|	+	+	+
x	-	-	|	+	+
x - 2	-	-	-	|	+
f(x)	-	+	-	+	+

By the First Derivative Test, f has a relative maximum at x = 0 and relative minima at x = - 1 and x = 2.