Question

Find all critical points of the function f(x) = x^3 + 5x^2 - 2x - 7. If there is more than one, list in descending order:The critical point(s) is(are) =

Answer 1

Given the function:

`f(x)=x^3+5x^2-2x-7`

The critical points are points where the derivative of the function is zero. So, first we find the derivate of the function:

`f^(\prime)(x)=3x^2+10x-2`

The derivative equals zero and we find the values for x:

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

We use the general formula for quadratic equations, where:

a = 3

b = 10

c = -2

`x=(-10\pm√(10^2-4(3)(-2)))/(2(3))`

Simplify:

`x=(-10\pm√(100+24))/(6)=(-10\pm√(124))/(6)`

Separate the solutions:

`\begin{gathered} x=(-10+√(124))/(6)=(-5+√(31))/(3) \\ and \\ x=(-10-√(124))/(6)=(-5-√(31))/(3) \end{gathered}`

Answer: the critical points are:

`x=(-5+31)/(3),(-5-31)/(3)`