Let the function f: R→ R be defined by { Question 1 (a) (b) f(x) = x2 x=0, x = 0. Explain the differentiability of f on R by using definition. Let f' be the first derivative of f computed in Q1(a). Explain why f' integrable on [1,2]. 7 Riemann
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- If I let f be continuously differentiable on R^2, how do I prove that gradf = (fx,fy)?Show that the function f(z)=sin x cosh y+i cosx sinh y is continious as well as analytic every where.Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) = 0. Use the given information and the Second Derivative Test to determine whether ƒ has a local minimum, a local maximum, or a saddle point at (0, 0), or state that the test is inconclusive. ƒxx(0, 0) = -9, ƒyy(0, 0) = -4, and ƒxy(0, 0) = -6
- Show that f(z) =xy+iy is everywhere continuous but is not analyticAssume that f : (0, 1) → R is a differentiable function with the derivative f' increasing on (0, 1). Prove that f' is continuous on (0, 1).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].