Problem 55 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(y'\).

Step 2: 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 \(x = -5\).

Step 3: 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 \(y = \frac{1}{2} * (-5)^{2} + 5 * -5 = -10\).