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Chapter 3: Q,20 (page 275)

For Exercises 15鈥�20, sketch the graph of a function f that has the indicated characteristics. If a graph is not possible, explain why.

f negative, f' positive, and f'' positive on 鈩�.

Short Answer

Expert verified

The possible graph of f is the following.

Step by step solution

01

Step 1. Given information. 

The given interval is 鈩�.

f is negative on interval.

f'is positive on interval.

f'' is positive on interval.

02

Step 2. sign chart of function. 

Suppose both f and f' are differentiable on an interval I.

(a) If f'' is positive on I, then f is concave up on I.

(b) If f'' is negative on I, then f is concave down on I.

So sign chart of functions follows.

sign chart state that the f is decreasing o interval.

Information of f'' did not properly state the graph of f.

antiderivative of f' is in the form of f(x)+c.

so we will do a vertical shift of f.

03

Step 3. Possible graph of functions. 

The possible graph off''follows.

The possible graph of f follows.

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Determine whether or not each function f in Exercises 53鈥�60 satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c 鈭� (a, b) that satisfy the conclusion of the Mean Value Theorem.

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Last night at 6 p.m., Linda got up from her blue easy chair. She did not return to her easy chair until she sat down again at 8 p.m. Let s(t) be the distance between Linda and her easy chair t minutes after 6 p.m. last night.

(a) Sketch a possible graph of s(t), and describe what Linda did between 6 p.m. and 8 p.m. according to your graph. (Questions to think about: Will Linda necessarily move in a continuous and differentiable way? What are good ranges for t and s?

(b) Use Rolle鈥檚 Theorem to show that at some point between 6 p.m. and 8 p.m., Linda鈥檚 velocity v(t) with respect to the easy chair was zero. Find such a place on the graph of s(t).
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