6. Theorem ( Interiar Extreme ) Let x. be interiar point of the interual I at which afunction f: I R has arelative extremun at x, If the derivative at x, exists , then f'(x) = 0 mathematical analysis
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- Fermat’s Theorem does not claim that if f'(c) = 0, then f (c) is a local extreme value (this is false). What does Fermat’s Theorem assert?Let f : I → J be a bijective differentiable function froman interval I to an interval J, and let F : I → R be an anti-derivative of f. Find an explicit expression, in terms of f, f-1 and F, for an anti-derivative of f-1: J → I. [Both substitution and integration by parts may come in handy.]The linearization of ex at x = 0 Derive the linear approximation ex = 1 + x at x = 0.
- 1. Show that f(t) is continuous at t=0 2. Use L'Hopital's rule to find the 1st and 2nd derivatives of f(t), evaluated at t =02 : Let F = (2x cos(x^2 + y^2), 2 + 2y cos(x^2 + y^2)) and C be the curve joining(-1,-1) to (-1,0) by traveling on the curves y = x^3 from (-1,-1) to (0,0), theny = x^2 from (0,0) to (1,1) and finally on y = − 1/4 (x + 1)(x − 3) from (1,1) to(-1,0). Evaluate∮C F · drwhat is the domain of f(x,y,z)=ln(z-y)+xysinz (with explaination please)
- A company has $300,000 to spend on two products. Suppose that the Utility derived by the company from x units of the first product and y units of the second product is given by U(x,y) = 3000x0.65y0.35 (i) Show that U is homogenous to some degree . (ii) Show that Euler’s Theorem holds for the function U(x,y)Consider a function f(x) = cos(x). calculate the exact value of the derivitave of f(x) at x=1.5. calculate the second order finite difference approximate value of the derivitave of f(x) at x=1.5, for delta-x=0.1.(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.