Math 22 Extrema and First Derivative Test

Relative Extrema

 Let  $f$  be a function defined at  $c$ .
 1.  $f(c)$  is a relative maximum of  $f$  when there exists an interval  $(a,b)$  containing  $c$  such that  $f(x)\leq f(c)$  for all  $x$  in  $(a,b)$ .
 2.  $f(c)$  is a relative minimum of  $f$  when there exists an interval  $(a,b)$  containing  $c$  such that  $f(x)\geq f(c)$  for all  $x$  in  $(a,b)$ .

If $f$ has a relative minimum or relative maximum at $x=c$ , then $c$ is a critical number of $f$ . That is, either $f'(c)=0$ or $f'(c)$ is undefined.

Relative extrema must occur at critical numbers as shown in picture below.

The First-Derivative Test

 Let  $f$  be continuous on the interval  $(a,b)$  in which  $c$  is the only critical number, then
 
 On the interval  $(a,b)$ , if  $f'(x)$  is negative to the left of  $x=c$  and positive to the right of  $x=c$ , then 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 a relative minimum.
 
 On 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 (a,b)}
, 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 f'(x)}
 is positive to the left 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 x=c}
 and negative to the right 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 x=c}
, then 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 a relative maximum.

Guidelines for Finding Relative Extrema

 1. Find the derivative 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}

 2. Find all critical numbers, then determine the test intervals
 3. Determine the sign 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'(x)}
 at an arbitrary number in each test intervals
 4. Apply the first derivative test

Exercises: Find all relative extrema of the functions below

1) 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)=x^2+2x}

Solution:
Step 1: 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)=2x+2=2(x+1)=0} ,
Step 2: Critical number is 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} , so the test intervals are 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,-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 (-1,\infty)}
Step 3: Choose 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 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,-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=0} 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 (-1,\infty)} .
Then we have: 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)=-2<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'(0)=2>0}
Step 4: 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 negative to the left 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 x=-1} and positive to the right of $x=-1$ , then $f(1)=3$ is a relative minimum
Therefore, Relative minimum: $(1,3)$
(Note: $f(x)$ in this case is a parabola so our answer makes sense)

2) $f(x)=6x^{\frac {2}{3}}+4x^{2}+3$

Solution:
Step 1: $f'(x)=6{\frac {2}{3}}x^{({\frac {2}{3}}-1)}+8x=4x^{\frac {-1}{3}}+8x=4x(x^{\frac {-4}{3}}+2)$ ,
Step 2: Critical number is $x=0$ , so the test intervals are $(-\infty ,-1)$ and $(-1,\infty )$
Step 3: Choose $x=-2$ for the interval $(-\infty ,-1)$ , and $x=0$ for the interval $(-1,\infty )$ .
Then we have: $f'(-2)=-2<0$ and $f'(0)=2>0$
Step 4: $f'(x)$ is negative to the left of $x=-1$ and positive to the right of $x=-1$ , then $f(1)=3$ is a relative minimum