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- Maximum divergence Within the cube {(x, y, z): | x | ≤ 1, | y | ≤ 1, | z | ≤ 1}, where does div F have the greatest magnitude when F = ⟨x2 - y2, xy2 z, 2xz⟩?1.Suppose that f : [0, 1] −→ [0, 1] is a continuous function. Prove that f has afixed point in [0, 1], i.e., there is at least one real number x ∈ [0, 1] such thatf(x) = x. 2.The axes of two right circular cylinders of radius a intersect at a right angle.Find the volume of the solid of intersection of the cylinders.true or false with reason if f is continuous in[a,b] then integrationba x f(x)dx= x integrationbaf(x)dx
- Let ƒ(x, y) =(x2-y2)/(x2+y2) for(x, y) ≠ (0, 0). Is it possible to define ƒ(0, 0) in a way that makes ƒ continuous at the origin? Why(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.Divergence theorem for F = 3 i + xy j + x k. Taken over a region bounded by z = 4 - y2, x = 0, x = 3, and the xy-plane.
- How do you find the extrema of a continuous function ƒ(x, y) on a closed bounded region of the xy-plane? Give an example.Let f: R^2->R be defined by f(x,y)= [(x^3+y^2)/(x^2+y^2) ,if (x,y)≠(0,0) 0, if (x,y)=(0,0).] Find if f is continuous at (0,0). Also find if f is differentiable at (0,0).Let f:Rto R be a continuous function that is convex on ( -infinity,0] and [0,infinity) and has a local maximum at the point 0 . Prove that the function is not differentiable at the point 0 .
- Supoose that g is continuous on [0,3] and differentiable on (0,3). Assume also that g has five zeros on [0,3]. What is the minimum number of critical numbers in the interval [0,3]. Support your answer.Find the degree of homogeneity of the function F(x, y) = x y/ x + y. Hence show that Euler′s Theorem holds?true or false with reason a)if f is continuous in[a,b] then integrationba x *f(x)dx= x integrationbaf(x)dx b) if f is continuous in[a,b] and f(x)>=0 then integrationbaf(x)1/2dx=(integrationba f(x)dx)1/2 c) If f and g are continuous and f(x) ≥ g(x) for a ≤ x ≤ b, then integrationbaf(x)dx>integrationbag(x)dx