A calculus problem by Alf-archie Sangkula
Calculus
Level
2
Given that f(x) + 2f(8-x) = x^2 for all real x, compute f(2)
67/4
76/2
70/3
68/3
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1 solution
Since we are looking for the value of f(2), let’s substitute x = 2 so that the first term becomes what we are looking for:
f(2) + 2f(6) = 4 \hspace{1em}-(1)
The term that is standing in our way to evaluate f(2) is f(6). Hence, it would be a good idea to find another relation with f(6) as one of the terms. Substituting x = 6 into the original functional equation:
f(6) + 2f(2) = 36 \hspace{1em}-(2)
How convenient! We now have 2 equations in f(2) and f(6), so we can solve for them.
2 \times (2) - (1): \hspace{1em} (2f(6) + 4f(2)) - (f(2) + 2f(6)) = 36 \times 2 - 4,
3f(2) = 68, f(2) = \displaystyle\frac{68}{3}.