Calculate the distance in $C[0,1]$ from the function $x(t) = t^2$ to the linear hull of the functions $y(t) = t$ and $z(t) = \sin t.$

Question

The following question is taken from 'Banach space Theory: The Basis for Linear and Nonlinear Analysis', Chapter $1,$ question $1.8.$

Question: Calculate the distance in $C[0,1]$ from the function $x(t) = t^2$ to the linear hull of the functions $y(t) = t$ and $z(t) = \sin t.$

So we are asked to calculate $$\inf_{a,b\in\mathbb{R}} \max_{t\in[0,1]}|t^2-at-b\sin(t)|.$$ Fix $a,b\in\mathbb{R}$ and denote $$f(t) = t^2-at-b\sin(t).$$ By differentiation, we obtain $$f'(t) = 2t-a-b\cos(t),$$ To find stationary point, we set $$f'(t) = 2t - a-b\cos(t)=0,$$ which implies that $$2t-a=b\cos(t).$$ But I have trouble solving the equation above.

Any hint would be appreciated.

Answer 1

You have to take the values at the endpoints $0$, $1$ into account as well. Anyway, I think this problem can only be solved numerically. By playing around with Mathematica I obtained the following approximation which shows that the distance in question is $\leq 0.026$.