How to find extreme values of a function.
Answers
Answer:
See below
Step-by-step explanation:
Extreme values of a function are found by taking the first derivative of the function and setting it equal to 0. To determine if it's a minimum or maximum, we set the second derivative equal to 0 and determine if its positive or negative respectively.
Let's do [tex]f(x)=3x^4+2x^3-5x^2+7[/tex] as an example
By using the power rule where [tex]\frac{d}{dx}(x^n)=nx^{n-1}[/tex], then [tex]f'(x)=12x^3+6x^2-10x[/tex]
Now set [tex]f'(x)=0[/tex] and solve for [tex]x[/tex]:
[tex]0=12x^3+6x^2-10x[/tex]
[tex]0=2x(6x^2+3x-5)[/tex]
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
[tex]x=\frac{-3\pm\sqrt{3^2-4(6)(-5)}}{2(6)}[/tex]
[tex]x=\frac{-3\pm\sqrt{9+120}}{12}[/tex]
[tex]x=\frac{-3\pm\sqrt{129}}{12}[/tex]
[tex]x=-\frac{3}{12}\pm\frac{\sqrt{129}}{12}[/tex]
[tex]x=-\frac{1}{4}\pm\frac{\sqrt{129}}{12}[/tex]
[tex]x_1=0,x_2\approx0.6965,x_3=-1.1965[/tex]
By plugging our critical points into [tex]f(x)[/tex], we can see that our extreme values are located at [tex](0,7)[/tex], [tex](0.6965,5.956)[/tex], and [tex](-1.1965,2.565)[/tex].
The second derivative would be [tex]f''(x)=36x^2+12x-10[/tex] and plugging in our critical points will tell us if they are minimums or maximums.
If [tex]f''(x)>0[/tex], it's a minimum, but if [tex]f''(x)<0[/tex], it's a maximum.
Since [tex]f''(0)=-10<0[/tex] then [tex](0,7)[/tex] is a local maximum
Since [tex]f''(0.6965)=15.822>0[/tex], then [tex](0.6965,5.956)[/tex] is a local minimum
Since [tex]f''(-1.1965)=27.18>0[/tex], then [tex](-1.1965,2.565)[/tex] is a global minimum
Therefore, the extreme values of [tex]f(x)=3x^4+2x^3-5x^2+7[/tex] are a global minimum of [tex](-1.1965,2.565)[/tex], a local minimum of [tex](0.6965,5.956)[/tex], and a local maximum of [tex](0,7)[/tex].
Hope this example helped you understand! I've attached a graph to help you visualize the extreme values and where they're located.
Related Questions
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b. What fraction of voters do not vote for the winning candidate? Write the fraction in simplest form.
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Answer:
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Step-by-step explanation:
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Answer:
m=fy/kx-Bf
Step-by-step explanation:
you simply simplify the equitions to be left with m on one side
Geometry!
Use the diagram of triangle MNO where, X, Y, and Z are the midpoints of the sides.
Answers
Answer:
YZ = 7 , MO = 8
Step-by-step explanation:
A segment joining the midpoints of 2 sides of a triangle is half the length of the third side.
(11)
YZ = [tex]\frac{1}{2}[/tex] MN , that is
3x + 1 = [tex]\frac{1}{2}[/tex] (10x - 6) = 5x - 3 ( subtract 5x from both sides )
- 2x + 1 = - 3 ( subtract 1 from both sides )
- 2x = - 4 ( divide both sides by - 2 )
x = 2
Then
YZ = 3x + 1 = 3(2) + 1 = 6 + 1 = 7
(12)
XY = [tex]\frac{1}{2}[/tex] MO , that is
x - 1 = [tex]\frac{1}{2}[/tex] (3x - 7) ← multiply through by 2 to clear the fraction
2x - 2 = 3x - 7 ( subtract 3x from both sides )
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which one is the same to 0.6 . 1/6 , 6% , 3/5 , 60% or 6/100
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Answer:
No because No one will brianliest me
Step-by-step explanation:
Answer:
I am alright. What about you?
The data for the distance d, in miles, remaining for a train to travel to its destination t hours after it departs a station are
shown in the scatter plot. Which of the following functions best fits the data?
Answers
Using the slope - intercept format, the equation which models the data represented by the graph is d(t) = -50t + 300
According to the general form of a linear regression equation :
y = bx + c b = slope ; c = interceptSlope, b = (Δy ÷ Δx)
Slope = (200 - 50) ÷ (2 - 5)
Slope = 150 ÷ -3 = -50
Intercept is the point where the regression line crosses the y - axis.
From the graph, the intercept = 300
Hence, the regression equation is y = -50t + 300
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Which is a solution for 7 (x plus 2) equals 35?
Answers
x = 5
f(x)=-x^2+9x+4f(x)=−x2+9x+4\text{Find }f(-8)Find f(−8)
Answers
Answer:
-74
Step-by-step explanation:
f(x) = -2x^2 +8x -10
Let x = -4
f(-4) = -2 ( -4)^2 +8(-4) -10
= -2(16) -32-10
= -32 -32-10
= -74
A painting is 15 in tall and 20 in wide. If it is reducted to a width of 4 in, then how tall will it be?
Answers
15/5=3
3in
(However, it is not specified whether it is also compressed in it’s height as well as it’s length - just stick with the 3 inch height)
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Answers
Answer:
c 3
Step-by-step explanation:
= -50880/800 = -318/5 x 20/20
= -6360/100
= -6360%
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Answers
Answer:
5.4% rounded to the tenths place.
Reasoning:
To convert 3/56 to a percentage, multiply the decimal version of 3/56 by 100.
Estimated, 3/56 as a percentage is 5.4%
I am a student in 7th honers and was wondering if someone can help me learn the basics of pre-algebra because I understand none of it.
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Answer:
I reccommend using an AoPS book. You can learn more at aops.com
Step-by-step explanation:
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Answer:
D is (-5,0)
E is (-1,2)
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Step-by-step explanation:
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How to solve algebraic inequalities.
Answers
Algebraic inequalities can be solved in the same way in which algebraic equations are solved.
Which expression is equivalent to 56+23m+12−16m?
Answers
Answer:
Step-by-step explanation:like like l'ike m y vieo9dsssssssss
The equivalent expression is 7m + 68
What is Equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the expression be A = 56 + 23m + 12 - 16m
Adding the like terms in the equation , we get
A = ( 56 + 12 ) + ( 23m - 16m )
On Simplifying , we get
A = 68 + 7m
A = 7m + 68
Hence , the required solution is 7m + 68
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Find the 66th term of the arithmetic sequence -10, 7, 24, ...−10,7,24,...
Answers
Answer:
a(66) = -10 + 17(65) = 1095
Step-by-step explanation:
We know that this is an arithmetic sequence. Subtracting -10 from 7 results in 17; subtracting 7 from 24 also results in 17. Thus, the common difference, d, is 17. The first term is -10.
The general form of an arithmetic sequence is a(n) = a(1) + d(n - 1).
Substituting -10 for a(1) and 17 for d, we get:
a(n) = -10 + 17(n - 1)
Now let n = 66: a(66) = -10 + 17(66 - 1), or a(66) = -10 + 17(65) = 1095
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Answer:
Step-by-step explanation:u have to add x+6 = ur answer there u go
If the ratio of grapes to walnuts to oranges is 6:6:12, how many grapes are there if the total number of fruits is 192?
Answers
Answer:
48 grapes,
Step-by-step explanation:
Firstly, let's sum up the ratio,
6+6+12=24
The fraction of grapes is therefore, 6÷24 or ¼
A ¼ of 192 is 48.
Signifying, there are 48 grapes.
A cheerilee teen willing to help,
stay positive, safe and Brainly!
If the ratio of grapes to walnuts to oranges is 6:6:12, how many grapes are there if the total number of fruits is 192?
Answer:-48 grapes
Explanation:-look at the attached picture :)
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a) 2
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that's the answer
Suppose you toss a fair coin​ 10,000 times. Should you expect to get exactly 5000​ heads? Why or why​ not? What does the law of large numbers tell you about the results you are likely to​ get?.
Answers
Answer:cottontail
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However, according to the law of large numbers, the frequencies of events with the same likelihood of occurrence even out, given enough trials or instances.
For example, in the case of a fair coin, where both head and tail have equal probability of occurrence, as the number of tosses becomes sufficiently large (say 1 million tosses), the ratio heads to tails in the outcome will be extremely close to 1:1.
So according to the law, we should expect to approach a point where half of the outcomes are heads and the other half are tails, as the number of tosses become very large.
Step-by-step explanation:
hope this helps
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Answers
Answer:
Perimeter is 16 cm
Step-by-step explanation:
3+5+2+6=16
im assuming the amounts given are the side lengths
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Answers
Step-by-step explanation:
x = 6
is a straight vertical line (parallel to the y-axis, going through x=6 on the x-axis).
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[tex]\\ \sf\longmapsto y=\dfrac{2}{7}x+\dfrac{1}{3}[/tex]
Compare to slope intercept form of a line
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Answer:
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Step-by-step explanation:
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he 134 sixth grade students sit together in the auditorium. Each row has 12 seats. How many rows do the sixth graders need
Answers
Answer:
Approximately 11.1666666667 rows
Step-by-step explanation:
134/12= 11.1666666667
Answer:
12 rows
Step-by-step explanation:
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The 6th-graders will need 12 rows of seats. The last row will not be filled.
A triangle has an area of 160 square inches and the base of the triangle is 40 inches. What is the height of the triangle?
Answers
Answer:
8 inches
Step-by-step explanation:
h = 2 A/b
A = 160
b = 40
160/40 = 4
2 x 4 = 8 inches
The height of the triangles is 8in.
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Find an ordered pair (x,y) that is a solution to the equation. 3x-y=8
Answers
Answer:
(2,-2)
Step-by-step explanation:
[tex]3x-y =3(2) -(-2)= 6 +2 = 8[/tex]
The lines represented by the equations 5y - x = 5 and y - x = 1 are
Answers
Answer:
Same line
Step-by-step explanation:
5y - x = 5 y - 1/5x = 1
5y = x + 5 y = 1/5x + 1
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y = 1/5x + 1