Question

Let C[0,1] gave inner product defined by < f,g >= ∫ f(x)g(x)dx ; integrate from 0 to 1

Evaluate d(f,g) where g(x) = c (constant) and f(x) = e^x. Use calculus to find the smallest value of E(c) = d^2(f, g) and call the value of c at which this occurs g*(x) = c*. Calculate the values d(f, g*) and also d(f, g) for the case g = (1 + e)/2. Comment (one sentence).