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Chapter 4: Problem 38

Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. $$f(x)=\frac{1}{15} x^{3}-x+1$$

Short Answer

Expert verified
Answer: The y-intercept for the function is located at (0,1), and the inflection point is at x = 0.

Step by step solution

01

Determine the Domain and Range

Since the given function is a cubic polynomial function, its domain will be all real numbers. There are no restrictions on the input values for x. For the range, the function has no restrictions either, as the cubic term will allow the function to cover all possible y-values. Therefore, the domain and range of this function are both all real numbers.
02

Find the Intercepts

To find the x-intercepts, we have to set the function f(x) equal to zero and solve for x: $$0=\frac{1}{15} x^{3}-x+1$$ This is a cubic equation and might be difficult to solve algebraically. We could use a graphing utility to find the approximate x-intercepts. To find the y-intercept, we can set x to 0: $$f(0)=\frac{1}{15}(0)^3 - (0) + 1 = 1$$ Therefore, the y-intercept is (0,1).
03

Identify Local Extreme Values (Critical Points)

We need to find the critical points by taking the first derivative of the function and setting it equal to zero: $$f'(x) = \frac{d}{dx}(\frac{1}{15} x^{3} - x + 1) = \frac{1}{5}x^2 - 1$$ Now we set f'(x) to zero and solve for x: $$\frac{1}{5}x^2 - 1 = 0$$ This yields the solutions: $$x = \pm\sqrt{5}$$ These are the critical points of the function. To determine whether these points are local maxima or minima, we can use the second derivative test.
04

Determine Inflection Points

We need to take the second derivative of the function and set it equal to zero: $$f''(x) = \frac{d^2}{dx^2}(\frac{1}{15} x^{3} - x + 1) = \frac{2}{5}x$$ Now we set f''(x) to zero and solve for x: $$\frac{2}{5}x = 0$$ This yields the solution: $$x = 0$$ This is the inflection point of the function since the concavity of the graph changes at this point.
05

Make a Complete Graph

Now, we can put all this information together to create a complete graph of the given function: - The domain and range of the function are both all real numbers. - The y-intercept is located at (0,1). - The x-intercepts can be determined using a graphing utility. - There are two critical points at $$x=\pm\sqrt{5}$$, and we can use the second derivative test to identify whether these points correspond to local maxima or minima. - There is an inflection point at x = 0. Using all this information, one can make a complete graph of the function \(f(x)=\frac{1}{15} x^{3}-x+1\).

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