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Chapter 4: Problem 21

Find the intervals on which \(f\) is increasing and decreasing. Superimpose the graphs of \(f\) and \(f^{\prime}\) to verify your work. $$f(x)=12+x-x^{2}$$

Short Answer

Expert verified
Answer: The function f(x) is increasing on the interval (-鈭�, 1/2) and decreasing on the interval (1/2, 鈭�).

Step by step solution

01

Find the first derivative f'(x)

First of all, we need to compute the derivative of the given function, which will give us information about its slope at any given point. The derivative of a polynomial is obtained by applying the following rule: $$\frac{d}{dx}(ax^n) = n * ax^{n-1}$$ So for our function: $$f(x) = 12 + x - x^2$$ We get the first derivative as: $$f'(x) = \frac{d}{dx} (12 + x - x^2) = 0 + 1 - 2x$$
02

Find the critical points of f'(x)

In order to find the intervals where the function is increasing or decreasing, we need to search for those points at which the first derivative is equal to zero or undefined (critical points). This will give us the boundaries between increasing and decreasing intervals. For our first derivative: $$f'(x) = 1 - 2x$$ This is a linear function and is defined for all values of x, so we should only look for the point at which f'(x) is equal to zero: $$1 - 2x = 0$$ Solving for x, we get: $$x = \frac{1}{2}$$
03

Analyze the sign of f'(x) on intervals around the critical point

Now, we have found the critical point x = 1/2. We will analyze the sign of the first derivative on the intervals determined by this critical point. 1. For x < 1/2: $$f'(x) = 1 - 2x > 0$$ Since the first derivative is positive, the function is increasing on the interval (-鈭�, 1/2). 2. For x > 1/2: $$f'(x) = 1 - 2x < 0$$ Since the first derivative is negative, the function is decreasing on the interval (1/2, 鈭�). So, the function f(x) is increasing on the interval (-鈭�, 1/2) and decreasing on the interval (1/2, 鈭�).
04

Graph of function and its first derivative

The last step is to superimpose the graphs of the function f(x) = 12 + x - x^2 and its first derivative f'(x) = 1 - 2x to verify our findings. When you plot these two functions, you will notice that the function is increasing where its derivative is positive, and decreasing where its derivative is negative, as we found through our calculations.

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