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Wize University Calculus 1 Textbook > Applications of Differentiation

Critical Points and Extrema

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Critical Points and Extrema

Since our first derivative can tell us intervals of increasing/decreasing, it can also give us locations of Extrema.

Critical Points

A critical point of a function f(x)f(x) is a point xx ,in its domain, for which f′(x)=0\boxed{f'(x)=0}

Singular Points

A singular point of a function f(x)f(x) is a point xx ,in its domain, for which f′(x)f'(x) does not exist.


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Relative Extrema

A function f(x)f(x) has a relative (local) maximum at a point xx, in its domain, if there is a small interval II , around xx, such that f(x)≥f(y) f(x)\ge f(y)\ for every yy in II.



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A function f(x)f\left(x\right) has a relative (local) minimum at a point xx, in its domain, if there is a small interval II, around xx, such that f(x)≤f(y)f(x)\le f(y)for every yy in II.


Wize Tip
Relative extrema occur at either critical points or singular points!

Example: Critical Points and Extrema

Find the critical points and extrema of f(x)=2x3−3x2−36x+2f(x)=2x^{3}-3x^{2}-36x+2.

Taking the derivative of the given function, we get

f′(x)=6x2−6x−36=6(x2−x−6)=6(x−3)(x+2)\begin{aligned} f'(x)=&6x^2-6x-36\\ =&6(x^2-x-6)\\ =&6(x-3)(x+2)\\ \end{aligned}



Critical points: set f′(x)=0⇒x=−2, x=3f'(x)=0 \Rightarrow x=-2,~x=3


Setting f′(x)=0f'(x)=0 implies that x=−2,x=3x=-2,x=3 which are the two values for f(x)f(x). Now, consider the following table

From the table above, we can conclude that the given function f(x)f(x) is increasing on (−∞,−2)(-\infty,-2) and (3,∞)(3,\infty) and decreasing on (−2,3)(-2,3). Therefore f(x)f(x) has a relative maximum at x=−2x=-2 and a relative minimum at x=3x=3.

Conclusion: x=−2x=-2 is relative maximum, x=3x=3 is relative minimum.

Find the critical points for f(x)=x−3x3\displaystyle f(x)=x-3\sqrt[3]{x}
Extra Practice
Critical points and extrema
Find all local extreme values of the function f(x)=x3−6x2+9x+1f(x)=x^3-6x^2+9x+1
Critical points and Extrema
For what values of pp the function f(x)=13x3+x2+px−1\displaystyle f(x)=\frac{1}{3}x^3+x^2+px-1 has at least one critical point.
Critical points and Extrema
Given this table of values, which of the following statements must NOT be true?
f(x)f′(x)f′′(x)x=0313x=100−1x=20−20\begin{array}{|c|c|c|c|} \hline &f(x)&f'(x)&f''(x)\\ \hline x=0&3&1&3\\ x=1&0&0&-1\\ x=2&0&-2&0\\ \hline \end{array}
Critical Points and Relative Extrema
Find all relative extrema for the function f(x)=4x−3x2/3f(x)=4x-3x^{2/3}.
Critical points and extrema
Where does the absolute maximum occur for the function f(x)=ex+1xf(x) = e^x + \frac{1}{x} on the interval [1,2][1, 2] .
Example: Absolute max and min
Example: Absolute max and minWize Concept Clarifier
Find the absolute extrema (max/min) of f(x)=e−x2+1f\left(x\right)=e^{-x^2}+1 on the interval [−1, ∞]\left[-1,\ \infty\right].
Relative Extrema
Find and classify all local and absolute extrema of f(x)=sin⁡xcos⁡xf\left(x\right)=\sin x\cos x on the interval [0,  2π]\left[0,\ \ 2\pi\right].
Max, Min, and Inflection Points
Given the following table of values, which statement(s) must always be true about f(x)f\left(x\right)? (Assume f(x)f\left(x\right) is continuous)
Select all that apply.
Critical Points and Extrema: Max and Min
Find and classify the critical points of f(x)=xex−exf(x)=xe^{x}-e^{x} as relative maxima or minima.
Critical Points and Extrema
What can be said about the extreme values of even and odd functions?
Q.\textbf{Q.} Find the absolute extrema of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2]. Then find the local/absolute extrema on its domain, and intervals of increase and decrease.
Critical Points and Relative Extrema
Find all relative extrema for the function f(x)=4x−3x2/3f(x)=4x-3x^{2/3}.
Continuity
Let f(x)f(x) be a continuous function on the open interval (a,b)(a, b). Which of the following four statements are always true.
Select all that apply.
Critical Points and Extrema
The following is the graph of f′′(x)f''\left(x\right), which of the following statements (if any) is correct?
Critical points and Extrema
Given this table of values, which of the following statements must NOT be true?
f(x)f′(x)f′′(x)x=0313x=100−1x=20−20\begin{array}{|c|c|c|c|} \hline &f(x)&f'(x)&f''(x)\\ \hline x=0&3&1&3\\ x=1&0&0&-1\\ x=2&0&-2&0\\ \hline \end{array}
Critical Points and Extrema
What can be said about the extreme values of even and odd functions?
What are the critical numbers for the function f(x)=x3exf(x)=x^3e^x ?
Conside the function f(x)=x1/4(x2−81)f(x)=x^{1/4}(x^2-81) which has derivative f′(x)=9(x2−9)4x3/4f'(x)=\frac{9(x^2-9)}{4x^{3/4}}. Which of the following lists all the critical points of f(x)f(x)?
Practice: Minimum Value (Similar to April 2018 Q45)
Find the minimum value of f(x)=14x4+13x3f(x)=\frac{1}{4}x^4+\frac{1}{3}x^3
How many critical numbers does the function f(x)=x2exf(x)=x^2e^x have?
Practice: Max, Min, and Inflection Points
Practice Question: Max, Min, and Inflection Points
Given the following table of values, which statement(s) must always be true about f(x)f\left(x\right)?
i.) f(x)f\left(x\right) has 3 critical points
Concavity, Inflection Points, Extrema and Intervals
Given f(x)=2−2x+13x3f(x)=2-2x+\frac{1}{3}x^3, find
the intervals of increase and decrease
the local maximum and minimum values
Critical points and extrema
Find all local extreme values of the function f(x)=x3−6x2+9x+1f(x)=x^3-6x^2+9x+1
Critical Points and Extrema
The graph of the derivative f′(x)f^{'}(x) is shown
Where are the local minimums of f(x)f(x)located?
The graph of the derivative f′(x)f^{'}(x) is shown
Where are the local minimums of f(x)f(x)located? Also list the intervals for which f is concave up or down.
Find the critical points for f(x)=x−3x3f(x)=x-3\sqrt[3]{x}.
The graph of the derivative f′(x)f^{'}(x) is shown
Where are the local minimums of f(x)f(x)located? Also list the intervals for which f is concave up or down.
Q.\textbf{Q.} Find the absolute extrema of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2]. Then find the local/absolute extrema on its domain, and intervals of increase and decrease.
Critical Points and Relative Extrema
Find all relative extrema for the function f(x)=4x−3x2/3f(x)=4x-3x^{2/3}.
Critical Points and Extrema: Max and Min
Find and classify the critical points of f(x)=xex−exf(x)=xe^{x}-e^{x} as relative maxima or minima.
Optimization
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
f(x)=x2−2ln⁡xf(x)=x^2-2\ln{x}
Critical points and Extrema
For what values of pp the function f(x)=13x3+x2+px−1\displaystyle f(x)=\frac{1}{3}x^3+x^2+px-1 has at least one critical point.
Consider the function f(x)=x2/3(x−5)f(x)=x^{2/3}(x-5) which has derivative f′(x)=5(x−2)3x1/3f'(x)=\frac{5(x-2)}{3x^{1/3}}. Which of the following lists all the critical points of f(x)f(x)?
Find the critical points for f(x)=x−3x3f(x)=x-3\sqrt[3]{x} as an ordered pair (x, y).
For the following function, determine all the local and global minimum/maximum and inflection points over the given interval
g(x)=x+2x on  (0,∞)g(x)=\sqrt{x}+\frac{2}{x} \text{ on }\ (0,\infty)
f′(x)f'(x) plot for a differentiable function is depicted below.
a) Find all critical point of ff in the range shown in the plot.
b) Determine which of the critical points found in the previous part are local max/min.
Find the critical points of f(x)=x+1x\displaystyle f(x)=x+\frac{1}{x}.
Find the critical point(s) of f(x)=x−3x3f(x)=x-3\sqrt[3]{x}.
Intervals of increase and decrease
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
f(x)=x3ln⁡xf(x) = x^3 \ln x
Optimization
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
y=e4x2xy=\frac{e^{4x}}{2x}
Which of the following statements is true about the function f(x)=(x2−3)e−xf\left(x\right)=\left(x^2-3\right)e^{-x}?
i.) The function has critical points at x=±3x=\pm\sqrt{3}
ii.) The function has a relative maximum at x=3x=3
Concavity and Inflection points
Give the graph of f′(x)f'\left(x\right) below, which of the following statments must be true about the graph of f(x)f\left(x\right)?
(Select all that apply)
Intervals of increase and decrease
If f′(x)=x5x+1(x−1)3(x+2)f'\left(x\right)=x^5\sqrt{x+1}\left(x-1\right)^3\left(x+2\right), which of the following statements about the graph of f(x)f\left(x\right) must be true?
(Select all that apply)
Example: Absolute max and min
Example: Absolute max and minWize Concept Clarifier
Find the absolute extrema (max/min) of f(x)=e−x2+1f\left(x\right)=e^{-x^2}+1 on the interval [−1, ∞]\left[-1,\ \infty\right].
Max, Min, and Inflection Points
Given the following table of values, which statement(s) must always be true about f(x)f\left(x\right)? (Assume f(x)f\left(x\right) is continuous)
Select all that apply.
Relative Extrema
Find and classify all local and absolute extrema of f(x)=sin⁡xcos⁡xf\left(x\right)=\sin x\cos x on the interval [0,  2π]\left[0,\ \ 2\pi\right].
Where does the absolute maximum occur for the function f(x)=ex+1xf(x) = e^x + \frac{1}{x} on the interval [1,2][1, 2] .
Critical points and extrema
Where does the absolute maximum occur for the function f(x)=ex+1xf(x) = e^x + \frac{1}{x} on the interval [1,2][1, 2] .
Critical Points and Extrema
Let f(x)f(x) be a smooth function (a function that has infinitely many derivatives that exist everywhere) and consider x=ax = a . Which of the following statements are correct? Check off all correct statements.
Critical points and extrema
f′(x)f'(x) plot for a differentiable function is depicted as below. Answer following questions.
Determine which of the critical points are local max/min
Critical points and extrema
f′(x)f'(x) plot for a differentiable function is depicted as below. Answer following questions.
Find all critical point of ff in the range shown in the plot.
Critical Points and Extrema
A math-inclined skier is going off a jump and getting his photo taken. His height above the photographer is given by f(t)=−t2+6t+5f(t)=-t^2+6t+5, where ttis the time in seconds after the skier leaves the ground. When should the photographer take the photo to get the skier at his maximum height above ground?
Critical points and extrema: Maximum and minimum on closed intervals
Find the absolute maximum value of f(x)=2ex(sin⁡x−cos⁡x)f(x)=2e^x(\sin x-\cos x)
on the interval [π/2,π][\pi/2,\pi].
Critical points and extrema: Maximum and Minimum on closed intervals
Find the absolute minimum of f(x)=arcsin(x2)f(x)=\text{arcsin}\left(\frac{x}{2}\right) on the interval [1,2][1,2].
Critical Points and Extrema: Maximum and Minimum on Closed Intervals
Find the absolute maximum of value of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1}
on the interval [0,2][0,2].