How can you determine where the tangent line is horizontal for the function y = 16x – x²?

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Finding the Points Where the Tangent Line is Horizontal

To find the points where the tangent line to the curve represented by the function y = 16x – x² is horizontal, we need to follow a few straightforward steps:

  1. Understand the concept of a tangent line:
    A tangent line is horizontal when its slope is zero. The slope of a curve at any given point can be found using derivatives.
  2. Differentiate the function:
    We start by calculating the derivative of the function. The derivative of y = 16x – x² is found using standard differentiation rules:

    y' = d/dx (16x) - d/dx (x²)
    = 16 - 2x
  3. Set the derivative equal to zero:
    To find where the tangent line is horizontal, we set the derivative equal to zero:

    16 - 2x = 0
  4. Solve for x:
    Rearranging the equation gives us:

    2x = 16  
    x = 8
  5. Find the corresponding y-coordinate:
    We now plug this value of x back into the original function to find the corresponding y-coordinate:

    y = 16(8) - (8)²  
    = 128 - 64  
    = 64
  6. Conclusion:
    The point at which the tangent line is horizontal is (8, 64).

In summary, the tangent line of the curve y = 16x – x² is horizontal at the point (8, 64). This step-by-step method demonstrates how to apply calculus principles to find critical points of interest on a graph.

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