Question

which functions have the property that the slope always increases as x increases?
select the correct choice below and, if necessary, fill in the answer box to complete your
choice.
a. (type a, b, c, or d. use a comma to separate answers as needed.)
b. none of the functions have the property that the slope always increases as x
increases.

Explanation:

Step1: Analyze the slope concept

The slope of a function \(y = f(x)\) is given by its derivative \(y'=f'(x)\). If the slope always increases as \(x\) increases, then the second - derivative \(y'' = f''(x)>0\) (since the derivative of the slope \(y'=f'(x)\) is \(y'' = f''(x)\), and if the slope is increasing, its rate of change is positive).

Step2: Analyze each graph

• For a function whose graph is concave - up, \(y''>0\).
• Graph A: It is a cubic function \(y = x^{3}\), \(y'=3x^{2}\), \(y'' = 6x\). The second - derivative \(y'' = 6x\) is not always positive (negative for \(x<0\)).
• Graph B: If we consider the general shape of the function (assuming it is a non - cubic, non - rational function with a single - valued \(y\) for each \(x\) in its domain), we can use the geometric interpretation of concavity. A concave - up function (where the slope of the tangent line increases as \(x\) increases) has \(y''>0\).
• Graph C: It is a rational function (has a vertical asymptote). Let's assume a general form \(y=\frac{1}{x - a}+b\). The first derivative \(y'=-\frac{1}{(x - a)^{2}}\) and the second derivative \(y''=\frac{2}{(x - a)^{3}}\). The sign of \(y''\) changes depending on the value of \(x\) relative to \(a\).
• Graph D: Similar to graph C (rational function with a vertical asymptote). Let \(y=\frac{m}{x - c}+d\). The first derivative \(y'=-\frac{m}{(x - c)^{2}}\) and the second derivative \(y''=\frac{2m}{(x - c)^{3}}\). The sign of \(y''\) is not always positive.

Answer:

B. None of the functions have the property that the slope always increases as \(x\) increases.