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Locating critical points a. Find the critical
Chapter 7, Problem 27E(choose chapter or problem)
23–30. Locating critical points
a. Find the critical points of the following functions on the domain or on the given interval.
b. Use a graphing utility to determine whether each critical point corresponds to a local minimum, local maximum, or neither.
\(f(x)=\left(e^{x}+e^{-x}\right) / 2\)
Questions & Answers
QUESTION:
23–30. Locating critical points
a. Find the critical points of the following functions on the domain or on the given interval.
b. Use a graphing utility to determine whether each critical point corresponds to a local minimum, local maximum, or neither.
\(f(x)=\left(e^{x}+e^{-x}\right) / 2\)
ANSWER:STEP_BY_STEP SOLUTION Step-1 Let f be a continuous function defined on an open interval containing a number ‘c’.The number ‘c’ is critical value ( or critical number ). If f (c) = 0 1 1 or f (c) is undefined. A critical point on that graph of f has the form (c,f(c)). Example ; Find all critical points of f(x) = x 8x . 2 Because f (x) is a polynomial function, its domain is all real numbers. 1 f (x) = 4x 16x f (x) = 0 =4x 16x 4x ( x - 4) = 0 4x(x+2)(x-2) = 0 , since a - b = (a+b)(a-b) X = 0 , x = -2 and x=2. 4 2 At x= -2 , then f(-2) = (2) 8(2) = 16 - 32 = -16. At x= 0 , then f(0) = (0) 8(0) = 0 - 0 = 0. At x= 2 , then f(2) = (2) 8(2) = 16 - 32 = -16. Hence,