A stock went on the market and began with a start value of $2. After 6 months of being on the market it reached its maximum value of 52$ but then dropped back down to 2$ by the 12th month. Three competing investment firms have come up with the following models.
Royal Bank: p(t) = 25/648 (t)2 (t-12)2+2
TD Investment: p(t) = -25 cos(pi/6 (t))+27
JerCO UltraSmart: p(t)= 50/log(7) ∙
log(t+1)+27
So my remaining questions given that info are.
1.For each case determine the intervals of time that would be good to buy the stock and the intervals or points that would be good to sell over the first 3 years on the market.
2.At an economics conference the stockbroker's investment subcommittee decides that Royal Bank's quartic function is good for short term and that JerCo UltraSmart's Is good for the long term. At what time value for "t" should they switch from the quartic to the logarithmic function?
3.The macro economics committee also looked at the data were convinced the stock data could be modelled by a cubic function. Show any and all calculations to determine a cubic function necessary shifts, stretches, or compressions.
1. To determine the intervals of time to buy and sell the stock for each case, we need to analyze the behavior of each function.
- Royal Bank: p(t) = (25/648)(t^2)(t-12)^2 + 2
To identify the intervals to buy, we look for the points where the function is at a minimum and increasing afterwards. To sell, we look for the points where the function is at a maximum and decreasing afterwards.
Starting with buying:
Since the function has a minimum at t = 12 (where the value is 2), we can consider buying the stock at or before this point.
For selling:
The function reaches its maximum at t = 6 months (where the value is 52). After this point, the function starts decreasing, so it would be ideal to sell before or at this point.
- TD Investment: p(t) = -25cos(pi/6(t)) + 27
To identify the intervals to buy, we need to find the points where the function is at a maximum (since the highest value indicates a good time to buy). To sell, we look for the points where the function is at a minimum.
Buying:
The maximum points of this function occur at t = 6 months and t = 18 months.
Selling:
The minimum point occurs at t = 12 months, which is the best time to sell.
- JerCO UltraSmart: p(t) = (50/log(7)) * log(t+1) + 27
To identify the intervals to buy, we look for the points where the function is increasing. To sell, we find the points where the function is decreasing.
Buying:
This function is always increasing, so any time within the first 3 years on the market would be suitable to buy.
Selling:
The function starts decreasing after t = 0, so any time after this point would be appropriate to sell.
2. To determine when to switch from the Royal Bank's quartic function to JerCO UltraSmart's logarithmic function, we need to find the time value "t" where their values intersect.
Setting the two functions equal to each other and solving for t:
(25/648)(t^2)(t-12)^2 + 2 = (50/log(7)) * log(t+1) + 27
This equation can be solved numerically using a graphing calculator or computer software.
3. To model the stock data using a cubic function, we need to find a suitable cubic equation that matches the given data points. Since we are given the start value ($2) and the maximum value ($52), we can use these points to determine a cubic function.
The general form of a cubic function is p(t) = at^3 + bt^2 + ct + d
We can set up a system of equations using the given data. Let's use the maximum value of $52 at t = 6 and the start value of $2 at t = 0.
52 = a(6^3) + b(6^2) + c(6) + d
2 = a(0^3) + b(0^2) + c(0) + d
Simplifying these equations gives:
216a + 36b + 6c + d = 52
d = 2
Substituting d = 2 into the first equation gives:
216a + 36b + 6c + 2 = 52
216a + 36b + 6c = 50
To have a unique solution, we would need another data point. With only two data points provided, we cannot uniquely determine the cubic function.