Section 4.3 Monotonic Functions and the First Derivative Test
Definition 4.3.1. Monotonic.
A function is monotonic on an interval if it is Definition 1.1.30 on that interval or decreasing on that interval.
Corollary 4.3.2.
Suppose \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\text{.}\)
If \(f'(x)\gt 0\) at every \(x\) in \((a,b)\text{,}\) then \(f\) is increasing on \([a,b]\text{.}\)
If \(f'(x)\lt 0\) at every \(x\) in \((a,b)\text{,}\) then \(f\) is decreasing on \([a,b]\text{.}\)
Proof.
Pick \(x_1\lt x_2\) in \([a,b]\text{.}\) The Mean Value Theorem guarantees that \(f(x_2)-f(x_1)=f'(c)(x_2-x_1)\) for some \(c\) between \(x_1\) and \(x_2\text{.}\) Since \(x_1\lt x_2\text{,}\) \(x_2-x_1\gt 0\text{,}\) so the sign of \(f'(c)\) is the same as the sign of \(f(x_2)-f(x_1)\text{.}\) So if \(f'(x)\gt 0\text{,}\) then \(f'(c)\gt 0\) and \(f(x_2)\gt f(x_1)\text{,}\) and if \(f'(x)\lt 0\text{,}\) then \(f'(c)\lt 0\) and \(f(x_2)\lt f(x_1)\text{.}\)
Example 4.3.3.
Find the critical points of \(f(x)=x^3+9x^2-2\) and then determine on which open intervals is \(f\) increasing and decreasing.
Suppose \(g\) is a function such that \(g'(x) = 3x^2(x+6)\text{.}\) Determine on which open intervals is \(g\) increasing and decreasing.
increasing on \((-\infty,-6)\) and \((0,\infty)\text{;}\) decreasing on \((-6,0)\)
increasing on \((-6,\infty)\text{;}\) decreasing on \((-\infty,-6)\)
Fact 4.3.4. First derivative test for local extrema.
Suppose \(c\) is a Definition 4.1.12 of a continuous function \(f\) and that \(f\) is differentiable on some open interval containing \(c\) except possibly at \(c\) itself. Moving from left to right:
If \(f'\) changes from positive to negative at \(c\text{,}\) then \(f\) has a local maximum at \(c\text{.}\)
If \(f'\) changes from negative to positive at \(c\text{,}\) then \(f\) has a local minimum at \(c\text{.}\)
If \(f'\) is the same sign on both sides of \(c\text{,}\) then \(f\) does not have an extremum at \(c\text{.}\)
Fact 4.3.5.
Absolute extrema can occur only at local extrema or endpoints of the domain. If the domain is unbounded, check \(\displaystyle\lim_{x\rightarrow\infty} f(x)\) and \(\displaystyle\lim_{x\rightarrow -\infty} f(x)\) to possibly identify local extrema as global extrema.
Example 4.3.6.
For each of the following: find the open intervals on which the function is increasing and decreasing, and find the function's local and absolute extrema, saying where they occur.
\(\displaystyle f(x) = -x^3+2x^2\)
\(\displaystyle g(x) = 2+4x-x^3-\frac14 x^4\)
\(\displaystyle h(x) = \frac{x^2-8}{x-3}\)
\(\displaystyle k(x)=\sqrt{81-x^2}\)
-
local minimum at \(x=0\)
local maximum at \(x=\frac43\)
-
absolute maximum at \(x=1\)
no minimum
-
local minimum at \(x=4\)
local maximum at \(x=2\)
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global maximum of 9 at \(x=0\)
global minimum of 0 at \(x=-9\) and \(x=9\)