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517 HW7 1. Let f : R → Rn (n ≥ 2) be differentiable with f 0 (t) 6= 0 for all t. Fix p ∈ / f (R) and let q be a point on f (R) with minimal distance to p (assumed to exist). Show that p − q is orthogonal to f (R) at q. 2. Suppose f : Rn → R is such that f (tx) = tf (x) for all t ∈ R and x ∈ Rn , but f is not linear. Show that f has directional derivatives at the origin, but is not differentiable there. Give an example of such a function. 3. Define f : R2 → R by f (x, y) = xy. Use the definition of derivative to show that f is differentiable everywhere, with df(a,b) (x, y) = bx + ay. 4. Let E ⊂ Rn be open and f : E → R differentiable. Prove that if f has a local maximum at a ∈ E, then f 0 (a) = 0. 5. For E ⊂ R2 , suppose f : E → R is differentiable with D1 f (x) = 0 for all x ∈ E. Under what condition on E can we say f depends on x2 only? 6. Define f : R2 → R by xy(x2 −y 2 ) , x2 +y 2 (x, y) 6= (0, 0) 0, (x, y) = (0, 0) ( f (x, y) = . Show that D1 f (0, y) = −y and D2 f (x, 0) = x for all x and y. Use this to show that D1 D2 f (0, 0) and D2 D1 f (0, 0) exist but are not equal. 1