To find the points where the tangent line is horizontal for the function y = 16x + 1 – x², we first need to determine where the derivative of the function is equal to zero. The derivative indicates the slope of the tangent line at any given point on the curve.
1. Differentiate the function: We start by differentiating the function with respect to x. The derivative of y is:
dy/dx = d(16x + 1 - x²)/dx = 16 - 2x
2. Set the derivative equal to zero: To find the points where the tangent line is horizontal, we set the derivative equal to zero:
16 - 2x = 0
3. Solve for x: Rearranging the equation, we get:
2x = 16
x = 8
4. Find the corresponding y value: Now, we substitute x = 8 back into the original equation to find the corresponding y-value:
y = 16(8) + 1 - (8)² = 128 + 1 - 64 = 65
5. Conclusion: Therefore, the point where the tangent line is horizontal is (8, 65).