Calculus

Find the x-coordinate(s) on the curve y = x - x^2 where the tangent line is horizontal.

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Understand what a horizontal tangent means

A horizontal tangent line has a slope of zero. Since the derivative dy/dx gives the slope of the tangent at any point, we need to find where dy/dx = 0.
2

Differentiate the curve y = x − x²

We differentiate term by term using the power rule. The derivative of x is 1, and the derivative of x² is 2x, so dy/dx = 1 − 2x.
3

Set the derivative equal to zero

We set the derivative equal to zero because that is where the slope of the tangent line is horizontal. Now we solve this equation for x.
4

Solve for x

Add 2x to both sides to get 2x = 1, then divide both sides by 2. There is exactly one solution: x = 1/2.

Understanding this problem

Learning Insight

The derivative of a function tells you the slope of the curve at every point. Setting it equal to zero finds the 'flat' spots — peaks, valleys, or saddle points. For a downward-opening parabola like this one, that flat spot is the single maximum point at the top of the arch.

Quick Tip

For any parabola y = ax² + bx + c, the horizontal tangent always occurs at x = −b/(2a), which is the vertex formula. Here a = −1, b = 1, so x = −1/(2·(−1)) = 1/2 — instant answer!

Common Mistake

Students often set the original equation y = x − x² equal to zero instead of the derivative. That finds the x-intercepts of the curve, not the point where the tangent is horizontal — two very different things.