Find an expression for a cubic function f if f(5) = 240 and
f(−3) = f(0) = f(6) = 0.
To find the expression for a cubic function, we need to first determine the general form of a cubic function, which is given as:
f(x) = ax^3 + bx^2 + cx + d
To find the specific values of a, b, c, and d for the given conditions, we can use the information provided:
1) f(5) = 240:
By substituting x = 5 into the equation, we get:
240 = a(5)^3 + b(5)^2 + c(5) + d
Simplifying this equation, we get:
240 = 125a + 25b + 5c + d
2) f(-3) = f(0) = f(6) = 0:
By substituting x = -3, x = 0, and x = 6 into the equation, we get three equations:
a(-3)^3 + b(-3)^2 + c(-3) + d = 0
a(0)^3 + b(0)^2 + c(0) + d = 0
a(6)^3 + b(6)^2 + c(6) + d = 0
Simplifying these equations, we get:
-27a + 9b - 3c + d = 0 Equation 1
d = 0 Equation 2
216a + 36b + 6c + d = 0 Equation 3
Now, we have a system of equations that we can solve to find the values of a, b, c, and d.
Using Equation 2, we know that d = 0.
Substituting this value into Equations 1 and 3, we get:
-27a + 9b - 3c = 0 Equation 4
216a + 36b + 6c = 0 Equation 5
To solve this system of equations, we can use any method available, such as substitution or elimination. Solving for a, b, and c, we find:
a = 1
b = -24
c = 36
Now, we can substitute these values into the general form of a cubic function:
f(x) = ax^3 + bx^2 + cx + d
f(x) = 1x^3 - 24x^2 + 36x + 0
f(x) = x^3 - 24x^2 + 36x
Therefore, the expression for the cubic function f, given the conditions f(5) = 240 and f(-3) = f(0) = f(6) = 0, is:
f(x) = x^3 - 24x^2 + 36x.