Question

Show that there are at least three distinct points x1,x2, and x3 such that f(x1)=f(x2)=f(x3)=10, where f(x)=x^3/(x^2-1)

Answer 1

Show that there are at least three distinct points $x1, x2$ , and $x3$ such that $f(x1) = f(x2) = f(x3) = 10$ , where $f(x) = x^3 / (x^2 - 1)$ .

Multiply 10 by each term inside the parentheses.

$$10x^2 - 10 = x^3$$

Since $x^3$ contains the variable to solve for, move it to the left-hand side of the equation by subtracting $x^3$ from both sides.

$$10x^2 - 10 - x^3 = 0$$

Move all terms not containing $x$ to the right-hand side of the equation.

$$-x^3 + 10x^2 - 10 = 0$$

Divide each term in the equation by -1.

$$x^3 - 10x^2 + 10 = 0$$

The solutions of the polynomial equation were found with the Durand-Kerner Method. There are 0 imaginary solutions.

$$x = 1.0575, -0.9554, 9.8979$$