2007
Problem - 4759
Let $f(c)=\int_0^1\left( (x-c)^2 + c^2\right)dx$ where $c$ is a real number. Find the minimal value of $f(c)$ as $c$ varies and the maximum value of $f(\sin\theta)$ as $\theta$ varies.
$f(c)$ can be calculated as $$f(c)=\left(\frac{1}{3}(x-c)^3 + c^2x\right)\bigg\vert_0^1=2c^2 - c + \frac{1}{3}=2\left(c-\frac{1}{4}\right)^2+\frac{5}{24}$$
Therefore, it has a minimum value of $\boxed{\frac{5}{24}}$ when $c=\frac{1}{4}$.
If $c=\sin\theta\in [-1, 1]$, then this quadratic function should have a maximum value when $c=-1$ because it is further from $\frac{1}{4}$. Accordingly, its maximum value equals $\boxed{\frac{10}{3}}$.