 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2023/6/c/solution

y = x (1 - x)

gives

J [y] = \int_{0}^{1} x^{3} (1 - x)^{2} d x = \frac{1}{60}, I [y] = \int_{0}^{1} x (1 - 2 x)^{2} d x + k^{2} \int_{0}^{1} x (1 - x)^{2} d x = \frac{1}{6} + \frac{k ^{2}}{12} .

5 k^{2} + 10

, yielding the upper bound. Conversely,

1/ x \geq x

(0, 1)

and the derivative energy is nonnegative, so

I [y] \geq k^{2} J [y]

for every admissible

y

. Part (b) therefore gives

k \leq z_{k, 1} \leq 5 k^{2} + 10 .

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