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Chapter 4: Problem 35

Find the intervals on which \(f\) is increasing and decreasing. $$f(x)=-2 x^{4}+x^{2}+10$$

Short Answer

Expert verified
Answer: The function is increasing on the intervals \((-\infty, -\frac{1}{2})\) and \((0, \frac{1}{2})\), and decreasing on the intervals \((-\frac{1}{2}, 0)\) and \((\frac{1}{2}, \infty)\).

Step by step solution

01

Find the derivative of the function

First, we will find the first derivative of the function \(f(x)\). This will help us identify the critical points at which the function changes from increasing to decreasing or vice versa. The function is: $$f(x)=-2 x^{4}+x^{2}+10$$ Using the power rule, the derivative is: $$f'(x) = -8 x^{3} + 2 x$$
02

Set the derivative to zero and find the critical points

To determine the critical points, we set the derivative equal to zero and solve for \(x\): $$-8 x^{3} + 2 x = 0$$ First, we can factor out a \(2x\): $$2x (-4 x^{2} + 1) = 0$$ Now, we can solve for \(x\): $$x=0$$ $$-4 x^{2} + 1 = 0$$ $$x^{2} = \frac{1}{4}$$ $$x = \pm\frac{1}{2}$$ Thus, we have obtained three critical points: \(x=-\frac{1}{2}, 0, \frac{1}{2}\).
03

Analyze the intervals between the critical points

We will now determine the sign of the first derivative on the intervals between the critical points. The intervals are: \((-\infty, -\frac{1}{2})\), \((-\frac{1}{2}, 0)\), \((0, \frac{1}{2})\), and \((\frac{1}{2}, \infty)\). For each interval, choose any point and evaluate the sign of the first derivative: Interval \((-\infty, -\frac{1}{2})\): Choose \(x=-1\) $$f'(-1) = -8(-1)^3 + 2(-1) = 6 > 0$$ Interval \((-\frac{1}{2}, 0)\): Choose \(x=-\frac{1}{4}\) $$f'(-\frac{1}{4}) = -8(-\frac{1}{4})^3 + 2(-\frac{1}{4}) = -\frac{1}{2} < 0$$ Interval \((0, \frac{1}{2})\): Choose \(x=\frac{1}{4}\) $$f'(\frac{1}{4}) = -8(\frac{1}{4})^3 + 2(\frac{1}{4}) = \frac{1}{2} > 0$$ Interval \((\frac{1}{2}, \infty)\): Choose \(x=1\) $$f'(1) = -8(1)^3 + 2(1) = -6 < 0$$
04

Identify the intervals where the function is increasing or decreasing

Based on the signs of the first derivative on the corresponding intervals, we can conclude the following: - The function is increasing on the intervals \((-\infty, -\frac{1}{2})\) and \((0, \frac{1}{2})\). - The function is decreasing on the intervals \((-\frac{1}{2}, 0)\) and \((\frac{1}{2}, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Points
Understanding critical points is central to analyzing a function's increasing and decreasing behavior. In calculus, critical points are values in the domain of a function where its derivative is zero or does not exist. Identifying these key points can help you pinpoint where a function may change from increasing to decreasing or vice versa.

For the function
$$f(x) = -2 x^{4} + x^{2} + 10$$
you identify critical points by setting its derivative, which you obtain using power rule differentiation, to zero and solving for the variable. In this case, you identified the critical points at
$$x=-\frac{1}{2}, 0, \frac{1}{2}$$
which are the x-values where the rate of change of the function momentarily stops鈥攎eaning there's neither an increase nor a decrease at exact those points.
First Derivative Test
The first derivative test is a valuable tool for determining if a critical point is a local maximum, local minimum, or neither. After identifying the critical points of a function, you must analyze the sign of the first derivative on intervals around these points. Here's how the test works:

Sign Analysis

For the given function, you divide the number line into intervals based on the critical points. Then, you test the sign of the derivative within each interval by substituting sample points into the derivative. The sign of the result tells you whether the function is increasing or decreasing on that interval.

Drawing Conclusions

If the first derivative changes from positive to negative at a critical point, the function has a local maximum there. On the other hand, if it changes from negative to positive, it indicates a local minimum. If there is no change, the test is inconclusive. For example, using the first derivative test on
$$f(x) = -2 x^{4} + x^{2} + 10$$
you found that the function is increasing where the first derivative is positive and decreasing where it is negative, thus, effectively visualizing the function's behavior around critical points.
Power Rule Differentiation
Power rule differentiation is a direct and simple way to find the derivative of functions with polynomials, where the derivative of each term is determined by bringing down the exponent as a coefficient and subtracting one from the exponent. Specifically, for a term
$$ax^n$$
its derivative is
$$n \times ax^{n-1}$$
In our problem's context, applying the power rule to
$$f(x) = -2 x^{4} + x^{2} + 10$$
gives us
$$f'(x) = -8 x^{3} + 2 x$$
This simplified process accelerates finding critical points and ultimately determining the function's increasing and decreasing intervals. Understanding the power rule sets a solid foundation for the study of calculus, as it is a critical skill for solving a broad range of problems involving derivatives.

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Most popular questions from this chapter

Is it possible? 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.

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=2 \cos t ; v(0)=1, s(0)=0$$

Even quartics Consider the quartic (fourth-degree) polynomial \(f(x)=x^{4}+b x^{2}+d\) consisting only of even-powered terms. a. Show that the graph of \(f\) is symmetric about the \(y\) -axis. b. Show that if \(b \geq 0\), then \(f\) has one critical point and no inflection points. c. Show that if \(b<0,\) then \(f\) has three critical points and two inflection points. Find the critical points and inflection points, and show that they alternate along the \(x\) -axis. Explain why one critical point is always \(x=0\) d. Prove that the number of distinct real roots of \(f\) depends on the values of the coefficients \(b\) and \(d,\) as shown in the figure. The curve that divides the plane is the parabola \(d=b^{2} / 4\) e. Find the number of real roots when \(b=0\) or \(d=0\) or \(d=b^{2} / 4\)

First Derivative Test is not exhaustive Sketch the graph of a (simple) nonconstant function \(f\) that has a local maximum at \(x=1,\) with \(f^{\prime}(1)=0,\) where \(f^{\prime}\) does not change sign from positive to negative as \(x\) increases through \(1 .\) Why can't the First Derivative Test be used to classify the critical point at \(x=1\) as a local maximum? How could the test be rephrased to account for such a critical point?

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=-32 ; v(0)=20, s(0)=0$$
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