f(x) is a function defined for all real values of x.
Which one of the following is a sufficient condition for ∫13 f(x) dx = 0?
- Af(2) = 0
- Bf(1) = f(3) = 0
- Cf(−x) = −f(x) for all x
- Df(x+2) = −f(2−x) for all x
- Ef(x−2) = −f(2−x) for all x
Show the answer and worked solution
answer · D
- Af(2) = 0
- Bf(1) = f(3) = 0
- Cf(−x) = −f(x) for all x
- Df(x+2) = −f(2−x) for all x
- Ef(x−2) = −f(2−x) for all x
An integral from 1 to 3 vanishes if the graph is antisymmetric about the midpoint x = 2, so look for the condition that says exactly that. Put u = x+2 in option D: then 2 − x = 4 − u, and the condition becomes f(u) = −f(4−u), which pairs each point with its mirror image in x = 2 and gives opposite values, so the two halves of the integral cancel. Options A and B fix the function at isolated points, which controls nothing. Options C and E both reduce to f being odd about the origin, which says nothing about the interval from 1 to 3.