Find all \( x \)-values where the line tangent to the curve \( F(x)=\left(x^{2}-14 x+24\right)^{700} \) is horizontal. Separate multiple answers with a comma. If there are no \( x \)-values where the curve has a horizontal tangent, write None for your answer.

To find the \( x \)-values where the line tangent to the curve \( F(x)=\left(x^{2}-14x+24\right)^{700} \) is horizontal, we need to find the values of \( x \) for which the derivative of the function is equal to zero. The derivative of \( F(x) \) with respect to \( x \) is given by: \[ F'(x) = 700\left(x^{2}-14x+24\right)^{699} \cdot (2x-14) \] To find the values of \( x \) where the tangent line is horizontal, we set the derivative equal to zero and solve for \( x \): \[ 700\left(x^{2}-14x+24\right)^{699} \cdot (2x-14) = 0 \] Solving this equation will give us the \( x \)-values where the tangent line is horizontal. Let's solve it. Solve the equation by following steps: - step0: Solve for \(x\): \(700\left(x^{2}-14x+24\right)^{699}\left(2x-14\right)=0\) - step1: Elimination the left coefficient: \(\left(x^{2}-14x+24\right)^{699}\left(2x-14\right)=0\) - step2: Separate into possible cases: \(\begin{align}&\left(x^{2}-14x+24\right)^{699}=0\\&2x-14=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=12\\&x=2\\&x=7\end{align}\) - step4: Rewrite: \(x_{1}=2,x_{2}=7,x_{3}=12\) The \( x \)-values where the line tangent to the curve \( F(x)=\left(x^{2}-14x+24\right)^{700} \) is horizontal are \( x = 2, 7, 12 \).