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# Prove: If A is invertible, then is invertible and

ISBN: 9780470432051 396

## Solution for problem 31 Chapter 2

Elementary Linear Algebra: Applications Version | 10th Edition

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Elementary Linear Algebra: Applications Version | 10th Edition

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Problem 31

Prove: If A is invertible, then is invertible and

Step-by-Step Solution:
Step 1 of 3

Minimum and maximum values Let c be a number in the domain of f. f(c) is a local max if f(c) ≥ f(x) when x is near c. f(c) is a local min if f(c) ≤ f(x) when x is near c Fermat’s Theorem: ​If f has a local max or min at c, and if f’(c) exists, then f’(c)’=0 Be careful: The converse of this theorem is not always true. Consider f(x) = x^3 f’(x) = 3x^2 f’(0) = 3*0^2 =0 However there is no min/ max at x=0 the tangent line is horizontal there. Consider f(x) = [x] F has a minimum at x = 0; however f’(0) does not exist. Consider f(x) = √x f has a minimum at x = 0 f’(x) = 1/√x f’(0) = 1/2 0 d oes not exist The tangent line is vertical there Def. a critical number of a function f is a number c in the domain of f such that either f’(c) = 0 or f’(c) does not exist Theorem: If a function has a min or max it occurs at a critical number of f 3 Ex. find all the critical numbers of f(x) = x√-x3​ Domain: all real numbers f(x) = x-3x^⅓ f’(x) = 1 - 3*1/3x^-⅔ = 1- 1/x^⅔ = 1 - 1/√x2 x = 0 is in the domain of f, but f’(0) does not exist f(x) = x - √​x so 0 is a critical number now set f’(x) equal to zero and solve 1 - 1/√x2 = 0 √3x2 *1 = 1/√x * √ x2 3 ( x ) ​3 = 1^3 x^2 = 1 x = -1, 1 So the critical numbers are -1, 0, 1 f(-1) = 2 this is a local max at x = 0 the tangent line is vertical but there is no max or min f(1) = -2 this is a local min Find the exact value of the minimum of f(x) = x^2ln(x) Domain x>0 f’(x) = x^2*(1/x)+2x ln(x) =x + 2x ln(x) Solve x + 2x ln(x) = 0 x

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##### ISBN: 9780470432051

The answer to “Prove: If A is invertible, then is invertible and” is broken down into a number of easy to follow steps, and 9 words. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. The full step-by-step solution to problem: 31 from chapter: 2 was answered by , our top Math solution expert on 03/13/18, 08:29PM. This full solution covers the following key subjects: . This expansive textbook survival guide covers 83 chapters, and 2248 solutions. Since the solution to 31 from 2 chapter was answered, more than 248 students have viewed the full step-by-step answer.

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