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Chapter 9: Problem 16

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{5}+1\)

Short Answer

Expert verified
The function \(y=x^{5}+ 1\) has no relative extrema and no points of inflection. The graph is always increasing and it looks like curve shifted one unit upwards from typical \(x^{5}\).

Step by step solution

01

Compute the First Derivative

The first derivative of the function \(y=x^{5}+1\) is found using the Power Rule of Derivatives. Hence the derivative \(y'\) can be calculated as \(y'=5x^{4}\)
02

Find the Critical Points

Set the first derivative equal to zero to find the critical points: \(5x^{4}=0\). The only solution to this equation is \(x=0\)
03

Compute the Second Derivative

Second derivative is derivative of first derivative which is: \(y''=20x^{3}\)
04

Identify the Inflection Points

To find inflection points, \(y''=20x^{3}=0\). The only solution to this equation is \(x=0\). However, because the concavity does not change sign, there are no points of inflection
05

Sketch the Graph of the Function

There are no relative extrema and no points of inflection, and the function is always increasing. You thus expect the graph to look like a curve that begins below the x-axis and is always increasing. It is shifted one unit above from typical \(x^{5}\) curve due to additional \(+1\) in original function

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Most popular questions from this chapter

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{1 / 3}+1\)

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{4}-4 x^{3}+16 x\)

Marginal Analysis, use differentials to approximate the change in cost, revenue, or profit corresponding to an increase in sales of one unit. For instance, in Exercise 29, approximate the change in cost as \(x\) increases from 12 units to 13 units. Then use a graphing utility to graph the function, and use the trace feature to verify your result. \(P=-x^{2}+60 x-100 \quad x=25\)

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{3}-4 x^{2}+6\)

The variable cost for the production of a calculator is \(\$ 14.25\) and the initial investment is \(\$ 110,000\). Find the total cost \(C\) as a function of \(x\), the number of units produced. Then use differentials to approximate the change in the cost for a one-unit increase in production when \(x=50,000\). Make a sketch showing \(d C\) and \(\Delta C\). Explain why \(d C=\Delta C\) in this problem.
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