Darboux's Theorem Prove Darboux’'s Theorem: Let f be differentiable on the closed interval [a, b] such that
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Chapter 3 Solutions
Calculus (MindTap Course List)
- Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity ∇g · n = Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)arrow_forward(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.arrow_forwardLet x and y be differentiable functions of t and lets = sqrt(x2 + y2) be the distance between the points (x, 0) and(0, y) in the xy-plane.a. How is ds/dt related to dx/dt if y is constant?b. How is ds/dt related to dx/dt and dy/dt if neither x nor y isconstant?c. How is dx/dt related to dy/dt if s is constant?arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage