Determine whether each of the following statements is true or false, and explain why in a few sentences1. A critical number c is a number in the domain of a function f for which f ′ ( c ) = 0 or f ′ ( c ) does not exist.2. If f ′ ( c ) > 0 on an interval, the function is positive on that interval.3. If c is a critical number,  then the function must have a relative maximum or minimum at c 4. If f ′ ( c ) exists, then f ″ ( c ) also exists5. If f ″ ( c ) > 0 on an interval, the function is increasing on that interval.

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Asked Nov 20, 2019
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Determine whether each of the following statements is true or false, and explain why in a few sentences

1. A critical number c is a number in the domain of a function f for which f ′ ( c ) = 0 or f ′ ( c ) does not exist.

2. If f ′ ( c ) > 0 on an interval, the function is positive on that interval.

3. If c is a critical number,  then the function must have a relative maximum or minimum at c

 4. If f ′ ( c ) exists, then f ″ ( c ) also exists

5. If f ″ ( c ) > 0 on an interval, the function is increasing on that interval.

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Expert Answer

Step 1

Since we are entitled to answer up to 3 sub-parts, we’ll answer the first 3 as you have not mentioned the subparts you need help with. Please resubmit the question and specify the other subparts you’d like to get answered.

Given:

  1. A critical number c is a number in the domain of a function f for which f ′ (c) = 0 or f ′ (c) does not exist.
  2. If f ′ (c) > 0 on an interval, the function is positive on that interval.
  3. If c is a critical number, then the function must have a relative maximum or minimum at c
Step 2

Solve (1):

A critical number c is the number in the domain of a function f for which f ′ (c) = 0 or f ′ (c) does not exist.

It is a true statement because it is the definition which is definition of critical number.

 A critical number of a function f is a point x = c in the domain of ‘f ‘such that either f ′(c) = 0 or f ′(c) does not exist.

Step 3

part (2):

If f ′ (c) > 0 on an interval, the function is positive on that interval.

It is the true statement by the definition of strictly inc...

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