Question

Find all the values of x where the tangent line to the function f(x)=x3-4x² - 2x + 7 is horizontal.The solution(s) is/are the value(s) of x that satisfy(Type an equation.)Solve for x.That is, solve the equationX=(Use a comma to separate answers as needed. Type an exact answer, using radicals as needed.)

Answer 1

Given the function f(x) expressed as

`f(x)=x^3-4x^2-2x+7`

For horizontal tangent, we have

`f^(\prime)(x)=0`

Thus, we have

`\begin{gathered} f^(\prime)(x)=3x^2-8x-2 \\ when\text{ f'\lparen x\rparen=0, we have} \\ 3x^2-8x-2=0 \\ \end{gathered}`

solving for x using quadratic formula, we have

`\begin{gathered} x=(-\left(-8\right)\pm\:2√(22))/(2\cdot\:3) \\ \Rightarrow x=(-\left(-8\right)+2√(22))/(2\cdot\:3) \\ =(4+√(22))/(3) \\ or \\ \Rightarrow x=(-(-8)2√(22))/(2*3) \\ =(4-√(22))/(3) \end{gathered}`

Hence, the solution are values of x that satisfy f'(x) = 0. That is, solve the equation

`3x^2-8x-2=0`

x=

`(4+√(22))/(3),(4-√(22))/(3)`