Final answer:
The statement is generally false because it assumes symmetry that isn't guaranteed by being a positive 1-1 function. Without knowing the symmetry of the function or the limits of integration, we cannot assure the integrals of xf(x) and (b-x)f(x) are equal.
Step-by-step explanation:
The question asks to evaluate the truth of a statement regarding integrals of a positive 1-1 function. To determine if the statement 'If f(x) is a positive 1−1 function, then π2π∫ab xf(x)dx=π2π∫ab (b-x)f(x)' is true or false, consider that for a 1-1 function, each x-value corresponds to a unique y-value and vice versa.
However, this statement about the integrals is generally false because without further information about the limits of integration or the symmetry of the function about the y-axis, we cannot guarantee the integral of xf(x) and the integral of (b-x)f(x) are the same.
When a function produces an odd function, such as xe-x² (odd times even is odd), it's true that the integral over all space of an odd function is zero. This is because the total area of the function above the x-axis cancels with the negative area below it. However, we cannot apply this property here without knowing if the integrand is an odd function.