Determine the value of “c” using the mean value theorem

Question

For the function $F(x) = Ax^2 + Bx + C$ determine the value of $c$ (critical point) at which the tangent line is parallel to the secant through the endpoints of the graph on the interval $[x1,x2]$. Not sure how to start this or do it at all so any help would be very much appreciated! Thank you in advance for your time and assistance!

Answer 1

According to the Mean Value Theorem: \begin{equation} F'(c) = \frac{Ax_{2}^2 + Bx_{2} + C - (Ax_{1}^2 + Bx_{1} + C)}{(x_{2} - x_{1})} \end{equation} \begin{equation} \implies F'(c) = \frac{A(x_{2}^2 - x_{1}^2) + B(x_{2} - x_{1})}{(x_{2} - x_{1})} \end{equation}

We can factor out $(x_2 - x_1)$, resulting in \begin{equation} F'(c) = A(x_{2} + x_{1}) + B \end{equation}

From this result, we can use the fact that $F'(x) = 2Ax + B$ and get: \begin{equation} 2Ac + B = A(x_{2} + x_{1}) + B \end{equation} \begin{equation} 2Ac = A(x_{2} + x_{1}) \end{equation} \begin{equation} c = \frac{(x_{2} + x_{1})}{2} \end{equation}

Hope it helps!!