4.5 Derivatives and the shape of a graph (Page 2/14)

Calculus volume 1 Page 6 / 14

Consider the function $f (x) = x^{3} - (\frac{3}{2}) x^{2} - 18 x .$ The points $c = 3, −2$ satisfy $f^{'} (c) = 0 .$ Use the second derivative test to determine whether $f$ has a local maximum or local minimum at those points.

$f$ has a local maximum at $−2$ and a local minimum at $3 .$

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We have now developed the tools we need to determine where a function is increasing and decreasing, as well as acquired an understanding of the basic shape of the graph. In the next section we discuss what happens to a function as $x \to ± \infty .$ At that point, we have enough tools to provide accurate graphs of a large variety of functions.

Key concepts

If $c$ is a critical point of $f$ and $f^{'} (x) > 0$ for $x < c$ and $f^{'} (x) < 0$ for $x > c,$ then $f$ has a local maximum at $c .$
If $c$ is a critical point of $f$ and $f^{'} (x) < 0$ for $x < c$ and $f^{'} (x) > 0$ for $x > c,$ then $f$ has a local minimum at $c .$
If $f ″ (x) > 0$ over an interval $I,$ then $f$ is concave up over $I .$
If $f ″ (x) < 0$ over an interval $I,$ then $f$ is concave down over $I .$
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 $f ″ (c) = 0,$ then evaluate $f^{'} (x)$ at a test point $x$ to the left of $c$ and a test point $x$ to the right of $c,$ to determine whether $f$ has a local extremum at $c .$

If $c$ is a critical point of $f (x),$ when is there no local maximum or minimum at $c ?$ Explain.

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For the function $y = x^{3},$ is $x = 0$ both an inflection point and a local maximum/minimum?

It is not a local maximum/minimum because $f^{'}$ does not change sign

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For the function $y = x^{3},$ is $x = 0$ an inflection point?

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Is it possible for a point $c$ to be both an inflection point and a local extrema of a twice differentiable function?

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Why do you need continuity for the first derivative test? Come up with an example.

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Explain whether a concave-down function has to cross $y = 0$ for some value of $x .$

False; for example, $y = \sqrt{x} .$

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Explain whether a polynomial of degree $2$ can have an inflection point.

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For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where $f$ is increasing or decreasing.

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−2, 0). Then it continues increasing a little before decreasing and crossing the x axis at (−1, 0). It achieves a local minimum at (1, −6) before increasing and crossing the x axis at (2, 0).

Increasing for $−2 < x < −1$ and $x > 2;$ decreasing for $x < −2$ and $−1 < x < 2$

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The function f’(x) is graphed. The function starts negative and touches the x axis at the origin. Then it decreases a little before increasing to cross the x axis at (1, 0) and continuing to increase.

Decreasing for $x < 1,$ increasing for $x > 1$

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The function f’(x) is graphed. The function starts positive and decreases to touch the x axis at (−1, 0). Then it increases to (0, 4.5) before decreasing to touch the x axis at (1, 0). Then the function increases.

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The function f’(x) is graphed. The function starts at (−2, 0), decreases to (−1.5, −1.5), increases to (−1, 0), and continues increasing before decreasing to the origin. Then the other side is symmetric: that is, the function increases and then decreases to pass through (1, 0). It continues decreasing to (1.5, −1.5), and then increase to (2, 0).

Decreasing for $−2 < x < −1$ and $1 < x < 2;$ increasing for $−1 < x < 1$ and $x < −2$ and $x > 2$

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For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where

$f$ is increasing and decreasing and
the minima and maxima are located.

The function f’(x) is graphed. The function starts at (−2, 0), decreases for a little and then increases to (−1, 0), continues increasing before decreasing to the origin, at which point it increases.

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The function f’(x) is graphed. The function starts at (−2, 0), increases and then decreases to (−1, 0), decreases and then increases to an inflection point at the origin. Then the function increases and decreases to cross (1, 0). It continues decreasing and then increases to (2, 0).

a. Increasing over $−2 < x < −1, 0 < x < 1, x > 2,$ decreasing over $x < −2,$ $−1 < x < 0, 1 < x < 2;$ b. maxima at $x = −1$ and $x = 1,$ minima at $x = −2$ and $x = 0$ and $x = 2$

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The function f’(x) is graphed from x = −2 to x = 2. It starts near zero at x = −2, but then increases rapidly and remains positive for the entire length of the graph.

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The function f’(x) is graphed. The function starts negative and crosses the x axis at the origin, which is an inflection point. Then it continues increasing.

a. Increasing over $x > 0,$ decreasing over $x < 0;$ b. Minimum at $x = 0$

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The function f’(x) is graphed. The function starts negative and crosses the x axis at (−1, 0). Then it continues increasing a little before decreasing and touching the x axis at the origin. It increases again and then decreases to (1, 0). Then it increases.

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For the following exercises, analyze the graphs of $f^{'},$ then list all inflection points and intervals $f$ that are concave up and concave down.

The function f’(x) is graphed. The function is linear and starts negative. It crosses the x axis at the origin.

Concave up on all $x,$ no inflection points

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The function f’(x) is graphed. It is an upward-facing parabola with 0 as its local minimum.

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The function f’(x) is graphed. The function resembles the graph of x3: that is, it starts negative and crosses the x axis at the origin. Then it continues increasing.

Concave up on all $x,$ no inflection points

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The function f’(x) is graphed. The function starts negative and crosses the x axis at (−0.5, 0). Then it continues increasing to (0, 1.5) before decreasing and touching the x axis at (1, 0). It then increases.

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The function f’(x) is graphed. The function starts negative and crosses the x axis at (−1, 0). Then it continues increasing to a local maximum at (0, 1), at which point it decreases and touches the x axis at (1, 0). It then increases.

Concave up for $x < 0$ and $x > 1,$ concave down for $0 < x < 1,$ inflection points at $x = 0$ and $x = 1$

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For the following exercises, draw a graph that satisfies the given specifications for the domain $x = [−3, 3] .$ The function does not have to be continuous or differentiable.

$f (x) > 0, f^{'} (x) > 0$ over $x > 1, −3 < x < 0, f^{'} (x) = 0$ over $0 < x < 1$

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$f^{'} (x) > 0$ over $x > 2, −3 < x < −1, f^{'} (x) < 0$ over $−1 < x < 2, f ″ (x) < 0$ for all $x$

Answers will vary

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$f ″ (x) < 0$ over $−1 < x < 1, f ″ (x) > 0, −3 < x < −1, 1 < x < 3,$ local maximum at $x = 0,$ local minima at $x = ± 2$

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There is a local maximum at $x = 2,$ local minimum at $x = 1,$ and the graph is neither concave up nor concave down.

Answers will vary

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Source: OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2

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