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Increasing and Decreasing Functions

The sign of the derivative tells you whether a function is rising or falling.

Reading the Sign of f'(x)

Example 1 — checking a simple parabola

f(x) = x^{2}, so f'(x) = 2 x . Describe where f is increasing and decreasing. 1 f'(x) > 0 when 2 x > 0, which means x > 0 2 f'(x) < 0 when 2 x < 0, which means x < 0 Answer: Increasing for x > 0 . Decreasing for x < 0 . This matches the usual U-shape of y = x^{2} .

Example 2 — a shifted parabola

f(x) = - x^{2} + 4 x, so f'(x) = - 2 x + 4 . Find where f is increasing. 1 Set up the inequality: f'(x) > 0 means - 2 x + 4 > 0 2 Rearrange: 4 > 2 x, so x < 2 Answer: Increasing for x < 2 (and by the same logic, decreasing for x > 2)

Practice

f(x) = x^{2} - 6 x + 5, so f'(x) = 2 x - 6 . Is f increasing or decreasing at x = 1 ?

More lessons in Calculus · Next: Stationary Points · Previous: Tangent Lines

Reading the Sign of f'(x)

Example 1 — checking a simple parabola

f(x) = x^{2}, so f'(x) = 2 x . Describe where f is increasing and decreasing. 1 f'(x) > 0 when 2 x > 0, which means x > 0 2 f'(x) < 0 when 2 x < 0, which means x < 0 Answer: Increasing for x > 0 . Decreasing for x < 0 . This matches the usual U-shape of y = x^{2} .

Example 2 — a shifted parabola

f(x) = - x^{2} + 4 x, so f'(x) = - 2 x + 4 . Find where f is increasing. 1 Set up the inequality: f'(x) > 0 means - 2 x + 4 > 0 2 Rearrange: 4 > 2 x, so x < 2 Answer: Increasing for x < 2 (and by the same logic, decreasing for x > 2)

Practice

f(x) = x^{2} - 6 x + 5, so f'(x) = 2 x - 6 . Is f increasing or decreasing at x = 1 ?