Question

[ y=2 x³-3 x²-72 x+17 \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The relative maximum is/are at {x}= ) (Type an integer

Answer 1

To find the relative maximum of the function y = 2x³ - 3x² - 72x + 17, we need to find the critical points and use the second derivative test. The relative maximum of the function y = 2x³ - 3x² - 72x + 17 is at x = -3.

Step-by-step explanation:

To find the relative maximum of the function y = 2x³ - 3x² - 72x + 17, we need to find the critical points where the slope of the function changes.

To find these points, we need to take the derivative of the function and set it equal to zero:

y' = 6x² - 6x - 72 = 0

Solving this equation, we find the critical points x = -3 and x = 6.

To determine if these points are relative maxima, we can use the second derivative test. The second derivative of the function is y'' = 12x - 6.

Substituting the critical points into the second derivative, we get:

y''(-3) = -42, which is negative, indicating a relative maximum at x = -3.

y''(6) = 66, which is positive, indicating no relative maximum at x = 6.

Therefore, the relative maximum of the function y = 2x³ - 3x² - 72x + 17 is at x = -3.