Question

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

Answer 1

$f\left(x\right)=x^3-x^2-x-\frac{11}{4}$

From graph, there is one local maxima and one local minima

$f\left(x\right)=3x^2-2x-1$

will have two real solutions

We have to chose k such that $f\left(x\right)=k$ have three real solutions

$f\left(x\right)=x^3-x^2-x-2.75=k$For $f\left(x\right)$ to have $3$ solutions, the last co efficient $-2.75$ needs to be positive and for that k must be equal to $-3$

$f\left(x\right)=x^3-x^2-x+0.25=k$