M1M1 Treasure Hunt

The M1M1 treasure hunt, week 7

Chapter 4: The Mean Valuation Theory

Ann was enjoying her weekly tutorials with the strange Doctor. Cheerfully, she skipped and danced towards the small blue box, not caring who was watching. But as she entered, a sombre Dr Hu greeted her. "It is time I dealt with the menace on the walkway. I wonder if you'd be willing to help. It might be dangerous."

"Dangerous? Look, I've been thinking about all this time travel stuff. How can you not know what's going to happen? All you have to do is travel further into the future and read the newspapers."
"Ah, yes, a popular fallacy. You are assuming there's only one universe. If you're not careful you can end up in the wrong one. There are lot of regulations to do with Time Travel. See that large book over there? Don't open it, because − "
" − it's much bigger on the inside, don't tell me," interrupted Ann.

"Indeed. But your question does relate to the matter in hand. It's to do with a dispute over Time Lord Pensions. You see, it's very easy to make money from time travel. All you need do is go back in time and invest a little money, return and pocket the dividends. But that is expressly forbidden, as such meddling can alter the past. However, as you suggest, you can go forward, see how the stock market is faring, come back and invest accordingly."
"And that doesn't count as meddling?"
"Only if done to excess. Which is precisely what the villain who has infiltrated the Blue Cube is doing. He travels around to different institutions, persuading everyone to undervalue their assets, buys them up, travels forward in time and releases them in the future's market. This is what caused the major bank collapse a few years ago. The Rules didn't permit me to prevent that, but now he's gone too far, undermining MY pension, which really must be stopped."

"But how does he persuade people to undervalue things?"
"He uses a very Mean Valuation Theory, which then powers his time machine. But we can foil his plans. The Pension fund will have a value t years in the future given by the function
P(t) = 7 t3-7t2+4t+10
The villain aims to cash in the fund in one year's time, at t = 1. But the real Mean Value Theorem guarantees that at some time over the next year the rate of increase of the fund will equal its average increase rate over the year. If I knew the t-value when this will occur I could travel to that time, and put the matter to rights."
t = ?

"Good. I also need to know the uniform growth rate which would minimise the mean-square-error. By which I mean I need to know the value of the constant A which minimises ∫01 (P(t)-At-P(0))2 dt."
Best A = ?

"Wonderful. I'll take us then, then. Now, professional ethics do not permit me get involved, but if anyone else were to enter the Pensions Office in the Blue Cube and type these numbers into the Control panel, the cube could only travel into the future at the normal rate. Effectively, it'd be immobilised." Handing her a scrap of paper covered with figures, he added "You should be safe. Well, fairly safe. Safe-ish. At least, I hope so. "
"So your professional ethics allow you to ask me to take risks, but not you?" demanded Ann scornfully.
"Indeed - risk minimisation is a very important investment principle. Oh, and the computer may be protected, but I'm sure you can work it out," replied the Doctor, flipping the controls in what was now a familiar manner.

Ann opened the door and looked out. She was inside a strange blue building. It was filled with many people, all apparently very busy, although it was not at all clear what, if anything, any of them were actually doing.
Boldly she entered a room marked "Pensions Office. No admittance except for unauthorised personnel." On the desk was an old-fashioned computer. When Ann pressed return, on the screen appeared:
"Security check. If at a point (x,y) on a curve dy/dx = 2, d2y/dx2 = 8 and d3y/dx3 = −480, then

d3x/dy3 = " ?

"Just as well I'm up-to-date with the problem sheets," thought Ann, as she obtained access to an incomprehensible spreadsheet, into which she entered Dr Hu's figures. To her relief, she was not disturbed. Once, a security guard came by, but satisfied that her presence was indeed unauthorised, he let her be.