Math 22 Concavity and the Second-Derivative Test

Formal Definition of Concavity

 Let $f$ be differentiable on an open interval $I$. The graph of $f$ is
 1. Concave upward on $I$ when $f'(x)$ is increasing on the interval.
 2. Concave downward on $I$ when $f'(x)$ is decreasing on the interval.

Test for Concavity

 Let $f$ be a function whose second derivative exists on an open interval $I$
 1. If $f''(x)>0$ for all $x$ in $I$, then the graph of $f$ is concave upward on $I$.
 2. If $f''(x)<0$ for all $x$ in $I$, then the graph of $f$ is concave downward on $I$.

Guidelines for Applying the Concavity Test

 1. Locate the $x$-values at which $f''(x)=0$ or $f''(x)$ is undefined.
 2. Use these $x$-values to determine the test intervals.
 3. Determine the sign of $f'(x)$ at an arbitrary number in each test intervals
 4. Apply the concavity test

Exercises: Find the second derivative of $f$ and discuss the concavity of its graph.

1) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=x^{3}+2x^{2}}

Solution:
Step 1: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)=3x^{2}+4x} , so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f''(x)=6x=0}
Step 2: So $x=0$, so the test intervals are $(-\infty ,0)$ and $(0,\infty )$
Step 3: Choose $x=-1$ for the interval $(-\infty ,0)$, and $x=1$ for the interval $(0,\infty )$.
Then we have: $f''(-1)=-6<0$ and $f''(1)=6>0$
Step 4: By the concavity test, $f(x)$ is concave up in $(0,\infty )$ and $f(x)$ is concave down in $(-\infty ,0)$

2) $f(x)=x^{4}-2x^{3}+10$

Solution:
Step 1: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)=4x^{3}-6x^{2}} , so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f''(x)=12x^{2}-12x=12x(x-1)=0}
Step 2: So, $x=0$ and $x=1$, so the test intervals are Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-\infty ,0),(0,1)} and $(1,\infty )$
Step 3: Choose $x=-1$ for the interval $(-\infty ,0)$, $x={\frac {1}{2}}$ for the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,1)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2} for the interval $(1,\infty )$.
Then we have: $f''(-1)=24>0$, $f''({\frac {1}{2}})=-3<0$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f''(2)=24>0}
Step 4: By the concavity test, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is concave up in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty,0)\cup (1,\infty)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is concave down in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,1)}

Points of Inflection

 If the graph of a continuous function has a tangent line at a point 
 where its concavity changes from upward to downward (or downward to upward), 
 then the point is a point of inflection.
 
 If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (c,f(c))}
 is a point of inflection of the graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}
, 
 then either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f''(c)=0}
 or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f''(c)}
 is undefined.

In exercises 1, at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0} , the concavity changes from concave down to concave up, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,f(0))} is a point of inflection.

Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,0)} is a point of inflection

In exercises 2, at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=1} , the concavity changes from concave up to concave down and from concave down to concave up, respectively. So, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,f(0))} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,f(1))} are points of inflection.

Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,10)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,9)} are points of inflection.

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