If a function f is continuous for all x and if f has a relative maximum at (-1, 4) and a relative minimum at (3, -2), which of the following statements must be true? (a) The graph of f has a point of inflection somewhere between x = -1 and x= 3
(b) f'(-1) = 0
(c) this is wrong
(d) The graph of f has a horizontal tangent line at x = 3
(e) The graph of f intersects both axes - Correct answer

I understand why e is correct, but I do not get why a, b, and d are all wrong. Aren't b and d to be expected since they are relative max/min's? I also can't imagine a case in which "a" is incorrect. Can someone explain why they are wrong? Thank you!

Answers

Answer 1

Answer:

First of all we need to review the concept of concavity. So, this is related to the second derivative. If we want to think about f double prime, then we need to think about how f prime changes, how the slopes of the tangent lines change.

So:

1) On intervals where $f''>0$ , the function is concave up (Depicted in bold purple in Figure 1)

2) On intervals where $f''<0$ , the function is concave down (Depicted in bold green in Figure 1)

Points where the graph of a function changes from concave up to concave down, or vice versa, are called inflection points.

Suppose we have a function whose graph is shown below. Therefore we have:

(a) The graph of f has a point of inflection somewhere between x = -1 and x= 3

This is true. As you can see from the Figure 2 the inflection point is pointed out in green. In this point the function changes from concave down to concave up.

(b) f'(-1) = 0

This is true. In $x=-1$ there's a maximum point. In this point the slope of the tangent line is in fact zero, that is, the function has an horizontal line as shown in Figure 3 (the line in green).

(c) this is wrong

This is false because we have demonstrated that the previous statement are true.

(d) The graph of f has a horizontal tangent line at x = 3

This is true. As in case (b) the function has an horizontal line as shown in Figure 3 (the line in orange) because in $x=3$ there is a minimum point.

(e) The graph of f intersects both axes

This is true according to Bolzano's Theorem. Apaticular case of the the Intermediate Value Theorem is the Bolzano's theorem. Suppose that $f(x)$ is a continuous function on a closed interval $[a,b]$ and takes the values of the opposite sign at the extremes, and there is at least one $c \in (a,b) \ such \ that \ f(c)=0$

Answer 2

Answer: $f'(x)=k(x+1)(x-3)$
$f'(x)=k(x^2-2x-3)$
$f(x)=k( (x^3)/(3)-x^2-3x )+C$
$f$
if f"(x)=0 then x=1
therefore a) is true
f'(-1)=0 b) is also true
f'(3)=0 d) is also true
e) option is also correct
i think

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5/12-3/20 use the least common multiple as the denominator

Answers

The common multiple of 12 and 20 is 60.
You've got to multiply the first one by 5 and the second by 3.
5*5= 25
This therefore becomes 25/60
The second becomes 3*3= 9
This becomes 9/60
As you're subtracting, you only subtract the numerators.
25-9= 16
Therefore, the answer is 16/60 which can be simplified to 4/10 or 2/5
Hope this helps :)

Kim bought a poster that cost $8.95 and a ome colored pencils.The total cost was $21.35.How much did the colored pencils cost?

Answers

if the total cost was 21.35, and the poster cost 8.95, then all you need to do is subtract to poster cost from the total cost to find the cost of the pencil.
21.35-8.95= $12.40

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Y= 3x - 10
y= 2x - 5

Answers

The solution to the system of equations is x = 5 and y = 5. The coordinates (5, 5) represent the point where the two lines intersect in the xy-plane, and it is the unique solution to this system of linear equations.

Given that the system of equations:

y = 3x - 10 (1)

y = 2x - 5 (2)

Since both equations are set equal to y, equate the right-hand sides of the equations since they represent the same value of y:

3x - 10 = 2x - 5

Now, let's solve for x:

3x - 2x = -5 + 10

x = 5

Now, found the value of x, substitute it back into either of the original equations to find the corresponding value of y.

Let's use the equation (1):

y = 3x - 10

y = 3(5) - 10

y = 15 - 10

y = 5

Hence, the values of x and y are 5 and 5. The system of equations have unique solution.

Learn more about Linear equation here:

brainly.com/question/32634451

#SPJ3

Complete question:

Solve the system of equations for x and y:

$y= 3x - 10\ny= 2x - 5$

If you're looking for a solution for both equations and had a choice between the 3 ways of solving it, I would choose substitution for this problem. So..

2x-5=3x-10

Move like terms to their own sides.

Take 2x away from both sides, and add 10 to both sides. That should leave you with

5=x

Now that you have the x, plug it into the equation that works best for you to solve.

That should be y=2(5)-5

y=10-5

y=5

(5,5) is your solution.

Point K is graphed on a number line at 3. Point L is 11 units away from point K on the number line.What could be the coordinates of point L?

Answers

3+/-11
add 11 to 3
11+3=14
minus 11 from 3
3-11=-8

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Answers

Ask yourself...

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And remember that 90% as a decimal is 0.9

===========

63 = 0.9x
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