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Chapter 9: Problem 6

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

Short Answer

Expert verified
The function \(y=x^{3}+3 x^{2}+3 x+2\) has a point of inflection and a relative maximum at the point \((-1,1)\).

Step by step solution

01

Find the first derivative

Differentiate the given function \(y = x^{3}+3x^{2}+3x+2\) with respect to \(x\). This will result in the first derivative \(y' = 3x^{2}+6x+3\). The critical points of the function are obtained by setting the first derivative equal to zero and solving for \(x\). Hence, \(3x^{2}+6x+3 = 0\).
02

Solve for critical points

Solving the equation \(3x^{2}+6x+3 = 0\) yields the critical points. You can factor the quadratic equation to find the roots, which are \(x = -1\).
03

Determine the critical points

Substitute the critical points into the original function to get the corresponding \(y\) -values. Hence, when \(x = -1\), \(y = -1 + 3 - 3 + 2 = 1\). So, the relative minimum or maximum occurs at the point \((-1,1)\).
04

Compute the second derivative

To classify the critical points determined in the previous step, compute the second derivative of the original function. This results in \(y'' = 6x+6\).
05

Classify the critical points

Use the critical points to determine whether the second derivative is positive or negative. If \(y''(-1)>0\), the critical point is a local minimum. If \(y''(-1)<0\), the critical point is a local maximum. Calculate \(y''(-1)\) to get -6, so \((-1,1)\) is a relative maximum.
06

Find the inflection points

The points of inflection of the graph can be determined by setting the second derivative equal to zero and solving for \(x\). Hence, \(6x+6 = 0\), which solves to give \(x = -1\). Substituting \(x = -1\) into the original function gives the point of inflection as \((-1,1)\).
07

Sketch the graph

Having determined the relative extrema and points of inflection of the function, proceed to sketch the graph. The graph has a point of inflection at \((-1,1)\) and a relative maximum at \((-1,1)\).

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Most popular questions from this chapter

The state game commission introduces 30 elk into a new state park. The population \(N\) of the herd is modeled by \(N=[10(3+4 t)] /(1+0.1 t)\) where \(t\) is the time in years. (a) Find the size of the herd after 5,10 , and 25 years. (b) According to this model, what is the limiting size of the herd as time progresses?

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