4.2: How Derivatives Affect the Shape of a Graph
Increasing/Decreasing Functions
A function is increasing on an interval $(a, b)$ if for any two numbers $x_1$ and $x_2$ in $(a, b)$, we have $f(x_1) < f(x_2)$ whenever $x_1 < x_2$.
A function is decreasing on an interval $(a, b)$ if for any two numbers $x_1, x_2 \in (a, b)$, we have $f(x_1) > f(x_2)$ whenever $x_1 < x_2$.
The derivative tells us exactly where a function is increasing and decreasing.
Increasing/Decreasing Test
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If $f'(x) > 0$ for $x \in (a, b)$ then $f$ is increasing on $(a, b)$.
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If $f'(x) < 0$ for $x \in (a, b)$ then $f$ is decreasing on $(a, b)$.
To use the I/D Test, first find critical numbers, then check signs of $f'(x)$ between critical numbers. This works since the derivative may change sign only when you move past a critical number.
Find intervals where $f(x) = x^2$ is increasing.
Determine the intervals where \[f(x) = x + \dfrac{1}{x}\] is increasing/decreasing.
Find where the function $f(x) = 3x^4 - 4x^3 - 12x^2 + 5$ is increasing or decreasing.
In the previous section we were concerned with finding absolute extrema. How about local extrema?
The First Derivative Test
Suppose $c$ is a critical number of a
continuous function $f$.
- If $f'$ changes from positive to negative at $c$, then $f$ has a local maximum at $c$.
- If $f'$ changes from negative to positive at $c$, then $f$ has a local minimum at $c$.
- If $f'$ changes doesn't change sign at $c$, then $f$ has no local maximum or minimum at $c$.
Using the First Derivative Test is the same method as the I/D Test. Just make sure the function is continuous.
Find the local minimum and local maximum values for the previous function.
Find the local minimum and local maximum values for the function $f(x) = x^4 - 4x^3$.
Concavity
The second derivative also tells us more information about a function.
A graph is concave up if all of the tangents lie below the graph. Alternatively, a graph is concave down if all of the tangents lie above the graph.
Concavity Test
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If $f''(x) > 0$ for all $x$ in an interval $I$, then the graph of $f$ is concave upward on $I$.
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If $f''(x) < 0$ for all $x$ in an interval $I$, then the graph of $f$ is concave downward on $I$.
Using the Concavity test is similar to the I/D Test.
- Find where $f''(x) = 0$ or $f''(x)$ does not exist (reason drawn in class).
- Create a sign diagram of $f''$.
Find the intervals of concavity for the function $f(x) = x^4 - 4x^3$.
Find the intervals of concavity for the function $f(x) = -x^4 + 2x^2 + 2$.
$f''(x)$ can also be used to find local minimums and maximums. The framework is similar to the First Derivative Test: find potential locations and test them.
Second Derivative Test
Suppose $f''(x)$ is continuous near $c$.
- If $f'(c) = 0$ and $f''(c) > 0$, then $f$ has a local minimum at $c$.
- If $f'(c) = 0$ and $f''(c) < 0$, then $f$ has a local maximum at $c$.
- If $f'(c) = 0$ and $f''(c) = 0$, the test is inconclusive.
Find all local minimums and maximums for the function $f(x) = x^4 - 4x^3$ using the Second Derivative Test.
Find all local minimums and maximums for the function $f(x) = -x^4 + 2x^2 + 2$ using the Second Derivative Test.