Exercise 2 Let f be the function defined on [0, 1] by if x + 0; {(2) = { sin(). if# 0; f(r) = 0, if r = 0. Prove that f is a Riemann integrable function.
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- (The Second Derivative Test) Let f : [a, b] → R be differentiable on (a, b). Suppose c ∈ (a, b) is such that f '(c) = 0, and f ''(c) exists. (a) If f ''(c) > 0, prove that f has a local minimum at c. (b) If f ''(c) < 0, prove that f has a local maximum at c. (c) Show, using two specific examples, that no conclusion can be made if f ''(c) = 0.what local linearizatoin of the function f(x,y) sin(xy) + cos(x/y) at (pie/4, 1) (show all steps please)Show that the Dirichlet function f defined on [0 , 1] by f(x) = {1 if x is rational0 if x is irrationalis not Riemann integrable on [0 , 1]
- Find the degree of homogeneity of the function F(x, y) = x y/ x + y. Hence show that Euler′s Theorem holds?f 1 (x) = x and f2 (x) = sin (x) sin Wronskian functions that are linearly independent show using.Show complete solution.Expand, that is, perform the indicated operations on the differential operator D.To expand this operator, apply it to a function y=f(x)1. xD(D + x)
- Let f(t) = et^2 be an integrable function defined on the closed interval [-x, x] on ℝ. If F is the anti-derivative of f on [-x, x], prove that F'(x) = f(x) for all x in [-x, x].Prove the identities assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar function, then div(f F) = f div(F) + F · ∇f .Apply the theorem of Cauchy-Riemann equations to verify that function f (z) is entire: f (z) = 3x + y + i(3y - x)