Problem 56
Question
Determine the point(s), if any, at which the graph of the function has a horizontal tangent line. $$ y=x^{2}+2 x $$
Step-by-Step Solution
Verified Answer
The point at which the graph of the function \(y = x^{2} + 2x\) has a horizontal tangent line is at (-1,-1).
1Step 1: Differentiate the function
First, differentiate the given function, \(y = x^{2} + 2x\). Using the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\), this gives the differentiated function as \(y' = 2x + 2\).
2Step 2: Equate the derivative to zero
Since the derivative represents the slope of the tangent line, and it is given that the slope is horizontal, meaning it has zero slope, equate the derivative to zero and solve for x, i.e. \(0 = 2x+2 -> x = -1\)
3Step 3: Find the corresponding y-coordinate
Substitute the value of x into the original function to find the corresponding y-coordinate. \(y = (-1)^2 + 2*(-1) = 1 - 2 = -1\). Therefore, the point of horizontal tangent is (-1,-1).
Other exercises in this chapter
Problem 56
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