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Chapter 3: Problem 10

Find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=2 x^{4}-8 x+3\)

Short Answer

Expert verified
The function does not have a point of inflection and it is concave upwards for all x in its domain.

Step by step solution

01

Find the derivative

The first step is to find the derivative of the function. The derivative of \(2x^4 - 8x + 3\) is \(f'(x) = 8 x^{3}-8\).
02

Find the second derivative

Next, find the second derivative, by finding the derivative of \(f'(x)\). The second derivative of \(f(x)\) i.e. \(f''(x)\) is \(24x^2\).
03

Find potential inflection points

The potential points of inflection are those values of \(x\) for which \(f''(x) = 0\). Solving \(f''(x) = 0\) yields \(x= 0\). There are no undefined points because \(f''(x)\) is a polynomial.
04

Test the Points

Now test the nature of the function and determine its concavity by analyzing the sign of the second derivative on either side of 0. If \(x < 0\), \(f''(x) > 0\) which means that function is concave upward on the interval \(-\infty,0\) and if \(x > 0\), \(f''(x) > 0\), which means that the function is also concave upward on the interval \(0,+\infty\). Since the concavity doesn't change, \(x=0\) is not an inflection point.

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Most popular questions from this chapter

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{x^{2}}{x^{2}-1} $$

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=2+\left(x^{2}-3\right) e^{-x} $$

Sketch the graph of \(f(x)=2-2 \sin x\) on the interval \([0, \pi / 2]\) (a) Find the distance from the origin to the \(y\) -intercept and the distance from the origin to the \(x\) -intercept. (b) Write the distance \(d\) from the origin to a point on the graph of \(f\) as a function of \(x\). Use a graphing utility to graph \(d\) and find the minimum distance. (c) Use calculus and the zero or root feature of a graphing utility to find the value of \(x\) that minimizes the function \(d\) on the interval \([0, \pi / 2]\). What is the minimum distance? (Submitted by Tim Chapell, Penn Valley Community College, Kansas City, MO.)

In Exercises 87 and \(88,\) (a) use a graphing utility to graph \(f\) and \(g\) in the same viewing window, (b) verify algebraically that \(f\) and \(g\) represent the same function, and (c) zoom out sufficiently far so that the graph appears as a line. What equation does this line appear to have? (Note that the points at which the function is not continuous are not readily seen when you zoom out.) $$ \begin{array}{l} f(x)=\frac{x^{3}-3 x^{2}+2}{x(x-3)} \\ g(x)=x+\frac{2}{x(x-3)} \end{array} $$

A line with slope \(m\) passes through the point (0,-2) . (a) Write the distance \(d\) between the line and the point (4,2) as a function of \(m\). (b) Use a graphing utility to graph the equation in part (a). (c) Find \(\lim _{m \rightarrow \infty} d(m)\) and \(\lim _{m \rightarrow-\infty} d(m)\). Interpret the results geometrically.
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