Question

PLS HELP
A certain stock's price, S (t), can be modeled by the polynomial function S (t) = t^4 – 12t^3 + 52t^2 – 96t + 64, where t represents the month in 2021. In which month will the price of the stock have a local maximum?

February
March
April
May

Answer 1

Answer:

April

Step-by-step explanation:

To solve, you find the maximum of the polynomial function. You can do this by either graphing or plugging in the values in a T chart.

I graphed the function, provided down below.

As you can see, the highest value in the graph is the middle most point, aka (4,1). Count down the months of the year to 4, and you'll find that April is the 4th month of the year.

Answer 2

Using derivative concepts, it is found that the function has a local maximum in March.

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• A function S(t) has local maximums or minimums at points in which the derivative is 0.
• If the function goes from increasing(positive derivative) to decreasing(negative derivative), it has a local maximum.

The function is:

$S(t) = t^4 - 12t^3 + 52t^2 - 96t + 64$

Which has the following derivative:

$S^{\prime}(t) = 4t^3 - 36t^2 + 104t - 96$

• Now, for each month, it has to be tested the derivative, verifying if it's zero.

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February is the 2nd month, thus, the derivative is:

$S^{\prime}(2) = 4(2)^3 - 36(2)^2 + 104(2) - 96 = 0$

Now, we have to test a value less than 2, and one greater than two, to check if it goes from increasing to decreasing, that is, derivative goes from positive to negative.

$S^{\prime}(1.5) = 4(1.5)^3 - 36(1.5)^2 + 104(1.5) - 96 = -7.5$

$S^{\prime}(2.5) = 4(2.5)^3 - 36(2.5)^2 + 104(2.5) - 96 = 1.5$

From negative to positive, thus, a local minimum in February.

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March is the 3rd month, thus, the derivative is:

$S^{\prime}(3) = 4(3)^3 - 36(3)^2 + 104(3) - 96 = 0$

Testing before and after:

$S^{\prime}(2.5) = 4(2.5)^3 - 36(2.5)^2 + 104(2.5) - 96 = 1.5$

$S^{\prime}(3.5) = 4(3.5)^3 - 36(3.5)^2 + 104(3.5) - 96 = -1.5$

From positive to negative, thus, it has a local maximum in March.

A similar problem is given at https://brainly.com/question/13539822