Problem 5. Determine whether the subspace of R(-1,1) consisting of all differentiable func- tions f (-1,1)→ R such that f'(0) = 0 is finite dimensional or not.

Q 2.8 Consider the functions f₁(x)=x and f₂(x)=3-5cx on the interval of [0,1]. so that a) Find the value of the constant c f, and £₂ are orthogonal on [0, 1]. Your answer should be symbolic and should not have any decimals ) Using the value of the constant c from part(a), find the norm of f₂ on the interval '[0, 1];

6.1.8

5. For n, m >0 and n + m, show T„(r)T (1) dr 0 VI- V1-r This is called the orthogonality relation for Chebyshev polynomials. Hint: Use T,(r) = cos(n cos-(x)) and z = cos e.