###### Calculus I

Exercices

Applications

Marginal analysis

# 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.

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