14. Let a € R" be a fixed vector. Define f(x) on R" by f(x) = a.x. Show f(x) is not coercive. Show that if & is any positive number, then the function g(x) = = a*x + & || x || ² is coercive.

Prove that (AXB)=D Ax dB dA XB np np np where A and B are differentiable functions of u.

Let ƒ(X) = ||X|, XER¹. For X 0 and e any unit vector, show that Vef(X) = (X, e) ||X||

Suppose f is a differentiable function on an interval II and f(x) 6= 0 for every x in the interval. Prove that f(x) and xf(x) are linearly independent on II.