III_Question1_Pure

(i) Looking at f'(\beta)=1+\frac{1}{\beta^2}+\frac{2}{\beta^3}, we see that the stationary point will be when \beta satisfies \beta^3+\beta+2=0. We see that this has a root at \beta=-1 (so we can factor it as (\beta+1)(\beta^2-\beta+2)).

To sketch the curve, we can use the fact that, for large (positive or negative) values of \beta, the two fractional terms will tend to zero, so f(\beta)\sim\beta. So, away from the y-axis, we will have something that resembles y=\beta. We know that it has a stationary point at \beta=-1, and we can see (by differentiating f'(\beta) again) that it is a local maximum. We also know that, towards \beta=0, the curve approaches -\infty. Combining this we can give a pretty good sketch of the curve. (Although, for the purposes of typing this up, I’ve cheated and not drawn it myself.)
Screenshot 2020-05-19 at 17.48.11

Asymptotically, we know that the curve y=g(\beta) will behave like that of y=f(\beta), and we need only worry about how it looks closer to \beta=0. Differentiating, we see that this has stationary points at \beta=-2 and \beta=1. Calculating g(\beta) at these two points lets us draw the graph, since we know where it has to turn around. (Here is a zoomed-in part of the graph.)
Screenshot 2020-05-19 at 17.51.40

(ii) We know that u and v are roots, which means that (x-u)(x-v)=x^2+\alpha x+\beta, whence u+v=-\alpha and uv=\beta. So u+v+\frac{1}{uv}=-\alpha+\frac{1}{\beta}.

With a bit more work, we can also calculate that \frac{1}{u}+\frac{1}{v}+uv = \frac{u+v}{uv}+uv = \frac{-\alpha}{\beta}+\beta.

(iii) By the previous part, we know that -\alpha+\frac{1}{\beta}=-1, which rearranges to give that \alpha=1+\frac{1}{\beta}. (N.B. if stuck, always remember that every previous part of the question will almost certainly be relevant! The choice of writing f(\beta) instead of f(x) in part (i) is also very suggestive). This means that \frac1u+\frac1v+uv = \frac{-\alpha}{\beta}+\beta = \beta-\frac1\beta-\frac1{\beta^2}. To show that this is \leqslant-1, we know (from part (i)) that it is enough to show that \beta\leqslant1 (by looking at our sketch and calculating f(1)).

Now note that we have yet to use the fact that u and v are real. This tells us that \alpha^2-4\beta\geqslant0, and, substituting \alpha=1+\frac1\beta, we see that -4\beta^3+\beta^2+2\beta+1\geqslant0. By inspection (and because we know that we’re interested in \beta=1) we see that this has \beta=1 as a root, and thus (1-\beta)(4\beta^2+3\beta+1)\geqslant0. To show that \beta\leqslant1 (which is equivalent to saying that 1-\beta\geqslant0) it thus suffices to show that 4\beta^2+3\beta+1\geqslant0. This can be done by e.g. completing the square (or just finding the minimum point, or some other method).

(iv) Inspired by the solution to part (iii), we know that we will probably do something similar, but using the second half of part (i) (the number 3 in both is a hint that this will be the case).

We are told (after using part (ii)) that \alpha=\frac1\beta-3. The fact that u and v are real again tells us (via the discriminant \alpha^2-4\beta\geqslant0) that -4\beta^3+9\beta^2-6\beta+1\geqslant0. This factors nicely as (1-\beta)^2(1-4\beta)\geqslant0. Since (1-\beta)^2 is always non-negative, we see that \beta must satisfy \beta\leqslant\frac14.

We can calculate that g(\frac14)=\frac{-15}{4}, which is useful, because, as before, \frac1u+\frac1v+uv=-\frac{\alpha}{\beta}+\beta=g(\beta) (finally using that \alpha=\frac1\beta-3). Looking at our graph from part (i), we know that the local maximum on the negative half of the \beta-axis is at \beta=-2, and we can calculate that g(-2)=\frac{-15}{4} as well. Since g is decreasing in the range 0\leqslant\beta\leqslant\frac14, and has a local maximum at \beta=-2, we know that the global maximum must be exactly \frac{-15}{4}.

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This solution is almost as impressive as the CleanSlate response speed!!! :fast_forward: :fast_forward: :fast_forward: :fast_forward: :fast_forward: