The function `f(x)=x^(3)-x^(2)-x-11/4` s graphed in the xy-plane above. If k is a constant such that the equation f(x)=k has three real solutions, which of the following could be the value of k ?

Options

A. 2
B. 0
C. `-2`
D. `-3`

Answer

D

Explanation

The equation f(x) = k gives the solutions to the system of equations `y = f(x) = x^3 − x^2 − x − 141` and y = k. A real solution of a system of two equations corresponds to a point of intersection of the graphs of the two equations in the xy-plane. The graph of y = k is a horizontal line that contains the point (0, k). Thus, the line with equation y = −3 is a horizontal line that intersects the graph of the cubic equation three times, and it follows that the equation f(x) = −3 has three real solutions.
Choices A, B, and C are incorrect because the graphs of the corresponding equations are horizontal lines that do not intersect the graph of the cubic equation three times.