Answer:
f(x) = (1/2)x^3 - 6x^2 + (45/2)x - 16
Step-by-step explanation:
Given: f(x) = ax^3 + bx^2 + cx + d
Both maximum and minimum satisfied the function, then:
f(3) = 11 => 11 = a(3)^3 + b(3)^2 + c(3) + d
11 = 27a + 9b + 3c + d (eq. 1)
f(5) = 9 => 9 = a(5)^3 + b(5)^2 + c(5) + d
9 = 125a + 25b + 5c +d (eq. 2)
The first derivative of the function is:
f'(x) = 3ax^2 + 2bx + c
And the second derivative is:
f''(x) = 6ax + 2b
In the relative minimum the first derivative is equal to zero, then:
f'(3) = 0 => 0 = 3a(3)^2 + 2b(3) + c
0 = 27a + 6b + c (eq. 3)
In the inflection point the second derivative is equal to zero, then:
f''(4) = 0 => 0 = 6a4 + 2b <=> b = -12a
Replacing b = -12a in eq. 3:
0 = 27a + 6(-12)a + c <=> c = 45a
Replacing b = -12a and c = 45a in eq. 1:
11 = 27a + 9(-12)a + 3(45)a + d
11 - 54a = d (eq. 4)
Replacing b = -12a and c = 45a in eq. 2:
9 = 125a + 25(-12)a + 5(45)a +d
9 - 50a = d (eq. 5)
Equating eq. 4 and eq. 5:
11 - 54a = 9 - 50a
2 = 4a
a = 1/2
So, b = -6, c = 45/2 and d = -16
