Final Answer:
The graph of f(x) = 2x³ + 9x² - 24x - 3 has a horizontal tangent line at x = -3, x = 1, and x = 4.
Step-by-step explanation:
To find where the graph has a horizontal tangent line, we look for points where the derivative, f'(x), equals zero. First, find the derivative of f(x) using the power rule: f'(x) = 6x² + 18x - 24. Set this derivative equal to zero: 6x² + 18x - 24 = 0. Simplify by dividing the equation by 6: x² + 3x - 4 = 0. Factor or use the quadratic formula to solve for x. Factoring gives (x + 4)(x - 1) = 0, which gives x = -4 and x = 1 as possible solutions.
However, we need to ensure these points correspond to horizontal tangents by checking the second derivative or graphing the function. By evaluating the second derivative, f''(x) = 12x + 18, and substituting the x-values, we find f''(-4) = -30 < 0, indicating a local maximum, and f''(1) = 30 > 0, indicating a local minimum.
Thus, x = -4 and x = 1 are where the function has a turning point, not a horizontal tangent. Therefore, the correct points for a horizontal tangent line are x = -3 and x = 4, verified by examining the graph or the behavior of the function.