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Chapter 4: Problem 36

Find the intervals on which \(f\) is increasing and decreasing. $$f(x)=\frac{x^{4}}{4}-\frac{8 x^{3}}{3}+\frac{15 x^{2}}{2}+8$$

Short Answer

Expert verified
Answer: The function is increasing on the intervals (0, 3) and (5, 鈭�), and decreasing on the intervals (-鈭�, 0) and (3, 5).

Step by step solution

01

Find the first derivative of the function

Compute the first derivative \(f'(x)\) with respect to \(x\) for the given function \(f(x)=\frac{x^{4}}{4}-\frac{8 x^{3}}{3}+\frac{15 x^{2}}{2}+8\). $$f'(x)=\frac{d}{dx}\left(\frac{x^{4}}{4}-\frac{8 x^{3}}{3}+\frac{15 x^{2}}{2}+8\right) = x^{3}-8x^{2}+15x$$
02

Find the critical points

Set the first derivative equal to zero and solve for the critical points. $$f'(x)= x^3 - 8x^2 + 15x = 0$$ Factor out \(x\): $$x(x^2-8x+15)=0$$ Factor the quadratic: $$x(x-3)(x-5)=0$$ Thus, the critical points are at \(x=0, x=3\), and \(x=5\).
03

Analyze the sign of the first derivative in each interval

Create a number line with the critical points and test each interval determined by the critical points: 1. Test the interval \((-\infty, 0)\): Choose \(x=-1\). \(f'(-1)=(-1)(-4)(-6)<0\) 2. Test the interval \((0, 3)\): Choose \(x=1\). \(f'(1)=1(-2)(-4)>0\) 3. Test the interval \((3, 5)\): Choose \(x=4\). \(f'(4)=4(1)(-1)<0\) 4. Test the interval \((5, \infty)\): Choose \(x=6\). \(f'(6)=6(3)(1)>0\)
04

Determine the intervals of increasing and decreasing

Based on the signs of the first derivative in each interval, we can determine if the function is increasing or decreasing: 1. \(f'(x)<0\) in the interval \((-\infty, 0)\), so \(f\) is decreasing. 2. \(f'(x)>0\) in the interval \((0, 3)\), so \(f\) is increasing. 3. \(f'(x)<0\) in the interval \((3, 5)\), so \(f\) is decreasing. 4. \(f'(x)>0\) in the interval \((5, \infty)\), so \(f\) is increasing. The function \(f\) is increasing on the intervals \((0,3)\) and \((5,\infty)\), and decreasing on the intervals \((-\infty,0)\) and \((3,5)\).

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Most popular questions from this chapter

The graph of \(f^{\prime}\) on the interval [-3,2] is shown in the figure. a. On what interval(s) is \(f\) increasing? Decreasing? b. Find the critical points of \(f .\) Which critical points correspond to local maxima? Local minima? Neither? c. At what point(s) does \(f\) have an inflection point? d. On what interval(s) is \(f\) concave up? Concave down? e. Sketch the graph of \(f^{\prime \prime}\) f. Sketch one possible graph of \(f\)

Differentials Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=1 / x^{3}$$

Determine whether the following properties can be satisfied by a function that is continuous on \((-\infty, \infty) .\) If such a function is possible, provide an example or a sketch of the function. If such a function is not possible, explain why. a. A function \(f\) is concave down and positive everywhere. b. A function \(f\) is increasing and concave down everywhere. c. A function \(f\) has exactly two local extrema and three inflection points. d. A function \(f\) has exactly four zeros and two local extrema.

Sketch the graph of a function that is continuous on \((-\infty, \infty)\) and satisfies the following sets of conditions. $$\begin{aligned}&f(-2)=f^{\prime \prime}(-1)=0 ; f^{\prime}\left(-\frac{3}{2}\right)=0 ; f(0)=f^{\prime}(0)=0\\\&f(1)=f^{\prime}(1)=0\end{aligned}$$

Use analytical methods to evaluate the following limits. $$\lim _{n \rightarrow \infty} \frac{1+2+\cdots+n}{n^{2}}( \text {Hint}:$$ $$\left.1+2+\cdots+n=\frac{n(n+1)}{2}.\right)$$
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