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Chapter 3: Q. 4 (page 274)

Sketch the graph of a function f that is concave down everywhere. Then draw five tangent lines on the graph, and explain how you can see that the derivative of f is decreasing.

Short Answer

Expert verified

The function f(x)=-2x4is zero for x=0and f(x)is concave down for all left and right ofx=0.

Step by step solution

01

Step 1. Given information

Function f is concave down everywhere

02

Step 2. derivative of the function.

If function f and its derivative f' is differentiable on an interval I and second derivative f'' is negative then f' is decreasing and f is concave down on interval I.

Consider a function f(x)=-2x4

first derivative,

f'(x)=-8x3

second derivative,

f''(x)=-24x2

f''(0)=0and f(x)''is negative for all left and right of x=0.

so f'(0)=0 and f'(x)is decreasing for all left and right of x=0.

f(0)=0and f(x)is concave down for all left and right of x=0.

03

Step 3. The graph of the function.

Plot the graph of the function f(x)=-2x4with five tangents.

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Most popular questions from this chapter

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).

Determine the graph of a function f from the graph of its derivative f'.

Find the possibility graph of its derivative f'.

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.

Use a sign chart for f'to determine the intervals on which each function fis increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

f(x)=cos2(x)
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