19. Properties of Graphs

c. First Derivative Information

1. Increasing & Decreasing

A function \(f(x)\) is increasing on an interval \([a,b]\) if for all \(x_1\) and \(x_2\) with \(a \le x_1 \lt x_2 \le b\), we have \[ f(x_1) \lt f(x_2) \] A function \(f(x)\) is decreasing on an interval \([a,b]\) if for all \(x_1\) and \(x_2\) with \(a \le x_1 \lt x_2 \le b\), we have \[ f(x_1) \gt f(x_2) \]

 Increasing        Decreasing

If \(f'(x) \gt 0\) for all \(x\) in \([a,b]\), then \(f(x)\) is increasing on the interval \([a,b]\).
If \(f'(x) \lt 0\) for all \(x\) in \([a,b]\), then \(f(x)\) is decreasing on the interval \([a,b]\).
Proof

So to find the intervals where \(f(x)\) is increasing and decreasing, we find the points where the derivative, \(f'(x)\), is \(0\) or undefined, plot them on a number line and test a point in each interval to find if \(f'(x)\) is positive or negative in that interval.

The critical points of a function \(f(x)\) are the values of \(x\) at which \(f'(x)\) is \(0\) or undefined.

Find the intervals where \(f(x)=\dfrac{x^4-39x^2-108}{3x}\) is increasing or decreasing.

To compute the derivative of \(f(x)\), we could use the Quotient Rule. However, it is easier to first simplify the function: \[\begin{aligned} f(x)&=\dfrac{1}{3}x^3-13x-\dfrac{36}{x} \\ f'(x)&=x^2-13+\dfrac{36}{x^2} \\ &=\dfrac{x^4-13x^2+36}{x^2}=\dfrac{(x^2-4)(x^2-9)}{x^2} \end{aligned}\] So the derivative is \(0\) at \(x=\pm2\) or \(x=\pm3\) and is undefined at \(x=0\). We plot these critical points on a number line: We test a point in each interval to see if \(f'(x)\) is positive or negative in each the \(6\) intervals. \[\begin{aligned} f'(-4)&=\dfrac{(16-4)(16-9)}{4^2} \gt 0 & f'(4)&=\dfrac{(16-4)(16-9)}{x^2} \gt 0 \\ f'(-2.5)&=\dfrac{(6.25-4)(6.25-9)}{2.5^2} \lt 0 \quad & f'(2.5)&=\dfrac{(6.25-4)(6.25-9)}{2.5^2} \lt 0 \\ f'(-1)&=\dfrac{(1-4)(1-9)}{1} \gt 0 & f'(1)&=\dfrac{(1-4)(1-9)}{1} \gt 0 \end{aligned}\] So \(f'(x)\) is positive on the intervals \((-\infty,-3)\), \((-2,0)\), \((0,2)\) and \((3,\infty)\), while it is negative on \((-3,-2)\), and \((2,3)\). We summarize these results on a number line: Based on the signs of the derivative, we indicate the intervals where \(f(x)\) is increasing and decreasing:

An isolated point \(x=p\) where \(f(p)=0\) but \(f'(x)\) does not change sign on the two sides is part of a single interval where \(f\) is increasing or decreasing, as in the next exercise. Such a point is called a horizontal inflection point as will be discussed more on the next three pages.