To determine the month in which the price of the stock will have a local maximum, we need to analyze the given polynomial function, S(t) = t^4 - 12t^3 + 52t^2 - 96t + 64.
In this case, t represents the month in 2021, and we want to find the month when the stock price reaches a local maximum.
A local maximum occurs when the value of the function is at its highest point within a specific interval. To find the local maximum, we need to identify the critical points of the function by finding where the derivative equals zero.
Let's find the derivative of the function S(t) with respect to t:
S'(t) = 4t^3 - 36t^2 + 104t - 96.
To find the critical points, we set the derivative equal to zero and solve for t:
4t^3 - 36t^2 + 104t - 96 = 0.
By factoring or using numerical methods, we find that one of the solutions is t = 3.
Now, we need to determine whether this critical point represents a local maximum or a local minimum. We can use the second derivative test to determine this.
Let's find the second derivative of S(t):
S''(t) = 12t^2 - 72t + 104.
Substituting t = 3 into the second derivative, we get:
S''(3) = 12(3)^2 - 72(3) + 104 = 40.
Since the second derivative is positive (S''(3) > 0), this indicates that the critical point at t = 3 represents a local minimum, not a local maximum.
Therefore, the price of the stock does not have a local maximum in February (t = 2), March (t = 3), or April (t = 4).
Hence, the correct answer is D) May.
If you have any further questions or need clarification, please let me know.