To find the points where the graph of the function f(x) = x² + x + 8 has a horizontal tangent line, we first need to calculate the derivative of the function. The derivative, f'(x), represents the slope of the tangent line at any point on the graph.
1. Calculate the derivative:
Using the power rule, we find:
f'(x) = 2x + 1
2. Set the derivative equal to zero:
To find the points where the tangent is horizontal, we set the derivative to zero:
2x + 1 = 0
3. Solve for x:
2x = -1
x = -1/2
4. Find the corresponding y-coordinate:
Now, we substitute x = -1/2 back into the original function to find the corresponding y-coordinate:
f(-1/2) = (-1/2)² + (-1/2) + 8
f(-1/2) = 1/4 – 1/2 + 8 = 1/4 – 2/4 + 32/4 = 31/4
5. Conclusion:
The point at which the graph has a horizontal tangent line is (-1/2, 31/4). This means that at this point, the slope of the tangent line to the graph is zero, indicating a flat or horizontal line.