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Chapter 7: Problem 66

Use a graphing utility to graph \(f\) and \(f^{\prime}\) over the given interval. Determine any points at which the graph of \(f\) has horizontal tangents. $$ f(x)=x^{3}-1.4 x^{2}-0.96 x+1.44 \quad[-2,2] $$

Short Answer

Expert verified
The solutions for \(x\) which are in the interval \([-2,2]\) are the \(x\)-coordinates of the points where \(f(x)\) has a horizontal tangent.

Step by step solution

01

Graph the function \(f(x)\)

First, a graphing utility can be used to plot the given function \(f(x) = x^{3} - 1.4x^{2} - 0.96x + 1.44\) over the specified interval of \([-2, 2]\). This will provide a visual representation of the function. Note the overall shape and any major points of interest.
02

Find the derivative \(f^{\prime}(x)\)

The derivative of the function, \(f^{\prime}(x)\), represents the slope of the tangent line at any point along the original function \(f(x)\). Use the power rule to take the derivative: for each term, multiply the coefficient by the current power of \(x\), and then decrease that power by 1. Doing so in this case gives: \(f^{\prime}(x) = 3x^2 - 2.8x - 0.96\).
03

Solve \(f^{\prime}(x) = 0\) for \(x\)

The derivative is equal to zero at points of horizontal tangency on the graph of \(f(x)\). Therefore, the values of \(x\) that solve the equation \(f^{\prime}(x) = 0\) represent the \(x \)-coordinates of this horizontal tangency. The roots of this equation \(3x^2 - 2.8x - 0.96 = 0\) can be found by applying the quadratic formula, \(x = [-b \pm \sqrt{b^2 - 4ac}]/(2a)\), where \(a = 3\), \(b = -2.8\) and \(c = -0.96\).
04

Check the solutions in the given interval

After obtaining the solutions for \(x\) from the quadratic formula, check to see if these \(x\) values are within the given interval \([-2, 2]\). If they are, they represent valid points of horizontal tangency.

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Most popular questions from this chapter

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{x+1}{x}} $$

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