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Chapter 3: Problem 24

Locate the absolute extrema of the function on the closed interval. $$f(x)=x^{3}-12 x,[0,4]$$

Short Answer

Expert verified
The absolute maximum of the function \( f(x) = x^{3} - 12x \) on the interval [0,4] is at \( x = 4 \) with value 16, while the absolute minimum is at \( x = 2 \) with value -16.

Step by step solution

01

Derive the function

Start by finding the derivative of the function \(f(x) = x^{3} - 12x\). Apply the power rule, which states that the derivative of \(x^n\) is \(n*x^{n-1}\), to both terms. This gives the derivative as \(f'(x) = 3x^2 - 12\).
02

Check critical points

Next, identify the critical points by setting the derivative equal to zero and solve for \(x\). So, \(3x^2 - 12 = 0\). Solving this equation gives \(x = -2\) and \(x = 2\). However, -2 is not within the given interval [0,4], so we only consider \(x = 2\). Additionally, note the endpoints of the interval, 0 and 4.
03

Evaluate the function

After obtaining the critical points and the endpoints, evaluate the function at these points. So, \(f(0) = 0^3 - 12*0 = 0\), \(f(2) = 2^3 - 12*2 = -16\) and \(f(4) = 4^3 - 12*4 = 16\).
04

Determine the absolute extrema

Now, obtain the absolute extrema by comparing these values. The highest value is called the absolute maximum and the lowest value is the absolute minimum. Here, the absolute maximum of the function on the interval [0,4] is \(f(4) = 16\) and the absolute minimum is \(f(2) = -16\).

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Most popular questions from this chapter

Maximum Volume A sector with central angle \(\theta\) is cut from a circle of radius 12 inches (see figure), and the edges of the sector are brought together to form a cone. Find the magnitude of \(\theta\) such that the volume of the cone is a maximum.

Use a graphing utility to graph the function and determine the slant asymptote of the graph. Zoom out repeatedly and describe how the graph on the display appears to change. Why does this occur? $$ f(x)=\frac{x^{3}}{x^{2}+1} $$

Sketch the graph of the function using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. $$ x^{2} y=4 $$

Investigation Consider the function \(f(x)=\frac{3 x^{n}}{x^{4}+1}\) for nonnegative integer values of \(n\). (a) Discuss the relationship between the value of \(n\) and the symmetry of the graph. (b) For which values of \(n\) will the \(x\) -axis be the horizontal asymptote? (c) For which value of \(n\) will \(y=3\) be the horizontal asymptote? (d) What is the asymptote of the graph when \(n=5\) ? (e) Use a graphing utility to graph \(f\) for the indicated values of \(n\) in the table. Use the graph to determine the number of extrema \(M\) and the number of inflection points \(N\) of the graph. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline n & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline M & & & & & & \\ \hline N & & & & & & \\ \hline \end{array} $$

Graphical Reasoning Consider the function \(f(x)=\tan (\sin \pi x)\) (a) Use a graphing utility to graph the function. (b) Identify any symmetry of the graph. (c) Is the function periodic? If so, what is the period? (d) Identify any extrema on \((-1,1)\). (e) Use a graphing utility to determine the concavity of the graph on \((0,1)\).
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