4.3 | Increasing and Decreasing Functions and the First Derivative Test

Test for Increasing and Decreasing Functions
Let $$f$$ be a function that is continuous on the closed interval $$[a,b]$$ and differentiable on the open interval $$(a,b)$$.

1. If $$f'(x) >0$$ for all $$x$$ in $$(a,b)$$, then $$f$$ is increasing on $$[a,b]$$.
2. If $$f'(x) < 0$$ for all $$x$$ in $$(a,b)$$, then $$f$$ is decreasing on $$[a,b]$$.
3. If $$f'(x) = 0$$ for all $$x$$ in $$(a,b)$$, then $$f$$ is constant on $$[a,b]$$.
The First Derivative Test
Let $$c$$ be a critical number of a function $$f$$ that is continuous on an open interval $$I$$ containing $$c$$. If $$f$$ is differentiable on the interval, except possibly at $$c$$, then $$f(c)$$ can be classified as follows.

1. If $$f'(x)$$ changes from negative to positive at $$c$$, then $$f$$ has a relative minimum at $$(c, f(c))$$.
2. If $$f'(x)$$ changes from positive to negative at $$c$$, then $$f$$ has a relative maximum at $$(c, f(c))$$.
3. If $$f'(x)$$ doesn’t change signs at $$c$$, then $$f(c)$$ is not a relative extrema.