Increasing/Decreasing & 1st Derivative Test

Unit 5 · Analytical Applications of Differentiation

Key Ideas

The sign of f′ tells you whether f is rising or falling. At a critical point where f′ changes sign, f has a local extremum.

f′(x) > 0 on (a, b) →

f \text{ is increasing on } (a,b)

f′(x) < 0 on (a, b) →

f \text{ is decreasing on } (a,b)

f′ changes − → + →

\text{local minimum at } c

f′ changes + → − →

\text{local maximum at } c

f′ no sign change →

\text{neither (e.g. inflection point)}

Evaluate f at:

\text{all critical points in } (a,b) \text{ and both endpoints } a,\, b

Then:

\text{largest value} = \text{abs. max}, \quad \text{smallest value} = \text{abs. min}

Type 1

Find the sign of f′ on each interval between critical points to determine where f is increasing (f′ > 0) or decreasing (f′ < 0).

Example 1

Find all intervals on which f is increasing and decreasing.

f(x) = x^3 - 3x^2 - 9x + 2

Practice more of this type— AI-generated · infinite problems

Type 2

At each critical point, check if f′ changes sign. Negative → positive means local minimum; positive → negative means local maximum.

Example 2

Find all relative maxima and minima of f.

f(x) = x^3 - 3x^2 - 9x + 2

Practice more of this type— AI-generated · infinite problems

Type 3

On a closed interval [a, b], the absolute max and min must occur at a critical point or at an endpoint. Evaluate f at all candidates and compare.

Example 3

Find the absolute maximum and minimum values of f on [−2, 4].

f(x) = x^3 - 3x^2 - 9x + 2, \quad [-2,\, 4]

Practice more of this type— AI-generated · infinite problems