Solution Found!
Explain why or why not Determine whether the
Chapter 7, Problem 47E(choose chapter or problem)
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function \(f(x)=\sqrt{x}\) has a local maximum on the interval [0, 1].
b. If a function has an absolute maximum, then the function must be continuous on a closed interval.
c. A function f has the property that f'(2) = 0. Therefore, f has a local maximum or minimum at x = 2.
d. Absolute extreme values on an interval always occur at a critical point or an endpoint of the interval.
e. A function f has the property that f'(3) does not exist. Therefore, x = 3 is a critical point of f.
Questions & Answers
QUESTION:
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function \(f(x)=\sqrt{x}\) has a local maximum on the interval [0, 1].
b. If a function has an absolute maximum, then the function must be continuous on a closed interval.
c. A function f has the property that f'(2) = 0. Therefore, f has a local maximum or minimum at x = 2.
d. Absolute extreme values on an interval always occur at a critical point or an endpoint of the interval.
e. A function f has the property that f'(3) does not exist. Therefore, x = 3 is a critical point of f.
ANSWER:STEP_BY_STEP SOLUTION Step-1 Critical point definition; 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)). Step-2 Absolute extreme values definition; When an output value of a function is a maximum or a minimum over the entire domain of the function, the value is called the absolute maximum or the absolute minimum. Let f be a f nction w ith domain D and let be a fixed constant in D . Then the output value f ) is the 1. Absolute maximum value of f on D if and only if f(x) f(c) , for all x in D. 2. Absolute minimum value of f on D if and only if f(c) f(x) , for all x in D. Step_3 Local maximum , local minimum definition; Let f be defined on the interval [a,b] , and x be the interior point on [a,b]. 0 A function f has a local maximum or relative maximum at a point x 0 if the values f(x) of f for x ‘near’ x are 0ll less than f(x ). 0 That is , f(x) f(x ) 0 Thus, the graph of f near x has a p eak at x . 0 0 A function f has a local minimum or relative