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Alik [6]
2 years ago
13

Determine the whether the quadratic equation below has a maximum or minimum. y = 4x² – 3x - 43

Mathematics
2 answers:
SashulF [63]2 years ago
8 0

Answer:

Minimum

Need to know:

Derivative power rule: d/dx xⁿ = nxⁿ⁻¹

Derivative constant rule: d/dx c = 0

Step-by-step explanation:

A way to determine whether the quadratic equation has a maximum or a minimum is by graphing it.

If we look at the image below, we notice that the graph does not go to the bottom infinitely, but it keeps going up infinitely. This means there is a minimum to this graph.

Another way to determine this is through calculus.

First, we have to find the point of extremum

Find the derivative of the given equation

d/dx 4x² -  3x - 43

You can separate the terms and solve them individually

d/dx 4x² = 4(2)(x²⁻¹) = 8x

d/dx -3x = -3(1)(x¹⁻¹) = -3(1)(1) = -3

d/dx -43 = 0

y' = 8x - 3

Set y' to equal 0

8x - 3 = 0

Add 3 to both sides

8x - 3 = 0

    + 3   + 3

8x = 3

Divide both sides by 8

8x/8 = 3/8

x = 3/8

There is only one point of extremum

Now we have to test whether the equation changes from negative to positive or positive to negative

If it changes from negative to positive, that point is a minimum

If it changes from positive to negative, that point is a maximum

To find this, pick a number less than 3/8 and plug it in the place of x in the derivative equation. We'll use 0.

y' = 8(0) - 3 = -3

Now we will do the same for a number larger than 3/8. We'll use 1

y' = 8(1) - 3 = 5

Since the lesser side of 3/8 is negative and the larger side of 3/8 is positive, that means it changes from negative to positive. This point is a minimum.

Alex2 years ago
4 0
A minimum because the function is positive. Positive quadratic functions always concave upwards (in a “bowl-like” shape), creating a curve/minimum at the bottom.
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2 years ago
The perimeter of a rectangle is 700 yards. What are the dimensions of the rectangle if the length is 50 yards more than the​ wid
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Answer:

width = 150 yards

length = 200 yards

Step-by-step explanation:

Using the information given to us, we can make the following two equations:

2L + 2W = 700

L = W + 50

where L is the length in yards and W is the width in yards. This could treat these two equations as a system of linear equations. There are multiple ways to solve a system of linear equations, but I am going to solve this by substitution. Before doing that, I am first going to simplify the first equation, as it will make solving the system easier.

2L + 2W = 700

Divide both sides by 2.

L + W = 350

Since L = W + 50, we can substitute W + 50 for L in the equation L + W = 700.

(W + 50) + W = 350

2W + 50 = 350

2W = 300

W = 150

Now that we found what W is, we can solve for L by plugging 150 into either one of the equations. I am going to plug it into the equation L = W + 50.

L = 150 + 50

L = 200

Now we found what L is. So now we know that the width is 150 yards and the length is 200 yards.

I hope you find my answer and explanation helpful. Happy studying. :)

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