Problem 4. For each of the following functions, prove whether or not it defines an inner product on the given vector space. (a) On R², (8.2) = ac- bd. (b) On P, (c) On Mnxn, (f(x),9(x)) = f* f'(x)9(x)dx. 0 (A, B) = tr(A+B).

If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?

= Evaluate [[2₁,22,23] ƒ (2) dz where 21 = −i, 22 = 2 + 5i, 23 5i and f(x + y) = x² + iy.

Suppose that u and v are functions, defined ontheir natural domains by the formuaeu(x) = √x and v(x) = x − x2Evaluate(v ◦ u) (4)