Question

Determine the point(s), if any, at which the graph of the function has a horizontal tangent line. $$ y=\frac{1}{2} x^{2}+5 x $$

Short answer

Therefore, the point at which the function has a horizontal tangent line is (-5, -10).

Step-by-step solution

Find the Derivative of the function

First, find the derivative of the function \(y=\frac{1}{2} x^{2}+5 x\). The power rule states that the derivative of \(x^{n}\) is \(n*x^{n-1}\). Following this rule, the derivative of \(y\) is \(y' = x + 5\). So, the slope of the tangent line to the graph of the function at any point is given by \(y'\).

Determine when the derivative equals zero

Next, in order to find the points at which this function has a horizontal tangent line, the derivative of the function must equal 0. So, we need to solve for \(x\) in the equation \(0 = x + 5\). Solving for \(x\), we obtain \(x = -5\).

Determine the corresponding y-coordinate

After finding the x-coordinate, we then calculate the corresponding y-coordinate by plugging \(x = -5\) into the original equation, resulting in \(y = \frac{1}{2} * (-5)^{2} + 5 * -5 = -10\).