Answer:
To find the value of f(5 we can set up and solve a system of equations using the given information.
Since f(x) is a polynomial of degree 4 we can write it in the general form: f(x) = ax^4 + bx^3 + cx^2 + dx + e.
Using the given values we can substitute the x values and their corresponding f(x) values into this equation:
f(1) = 1: a(1)^4 + b(1)^3 + c(1)^2 + d(1) + e = 1
f(2) = 2: a(2)^4 + b(2)^3 + c(2)^2 + d(2) + e = 2
f(3) = 3: a(3)^4 + b(3)^3 + c(3)^2 + d(3) + e = 3
f(4) = 4: a(4)^4 + b(4)^3 + c(4)^2 + d(4) + e = 4
f(0) = 1: a(0)^4 + b(0)^3 + c(0)^2 + d(0) + e = 1
Simplifying these equations gives us the system:
a + b + c + d + e = 1
16a + 8b + 4c + 2d + e = 2
81a + 27b + 9c + 3d + e = 3
256a + 64b + 16c + 4d + e = 4
e = 1
We can solve this system to find the values of a b c d and e. Then we can substitute x = 5 into the equation f(x) = ax^4 + bx^3 + cx^2 + dx + e to find f(5).
Solving this system of equations we find:
a = -1/12
b = 7/24
c = -7/12
d = 11/24
e = 1
Substituting x = 5 into the equation f(x we get:
f(5) = (-1/12)(5)^4 + (7/24)(5)^3 + (-7/12)(5)^2 + (11/24)(5) + 1
Evaluating this expression we find that f(5) = 101/12.
Therefore f(5) = 8.41667.