The cubic function with the desired properties is:
f(x) = (-3/4)x³ - (9/4)x² - (113/4)x + 63/4
To find the values of a, b, c, and d, we can use the given information about the relative maximum, relative minimum, and point of inflection.
Relative Maximum:
The point (-7, 163) is a relative maximum. At this point, the derivative of the cubic function is equal to zero. Taking the derivative of the cubic function, we have:
f'(x) = 3ax² + 2bx + c
Setting x = -7 and f'(-7) = 0, we get:
49a - 14b + c = 0
Relative Minimum:
The point (5, -125) is a relative minimum. At this point, the derivative of the cubic function is equal to zero. Taking the derivative of the cubic function, we have:
f'(x) = 3ax² + 2bx + c
Setting x = 5 and f'(5) = 0, we get:
75a + 10b + c = 0
Point of Inflection:
The point (-1, 19) is a point of inflection. At this point, the second derivative of the cubic function changes sign. Taking the second derivative of the cubic function, we have:
f''(x) = 6ax + 2b
Setting x = -1, we get:
-6a + 2b = 0
Solving the system of equations formed by the above three equations, we can find the values of a, b, c, and d.
49a - 14b + c = 0
75a + 10b + c = 0
-6a + 2b = 0
Solving these equations, we find:
a = -3/4
b = -9/4
c = -113/4
d = 63/4
Therefore, the cubic function with the desired properties is:
f(x) = (-3/4)x³ - (9/4)x² - (113/4)x + 63/4