Question

If `f(x)` is an even function and satisfies the relation `x^2dotf(x)-2f(1/x)=g(x),w h e r eg(x)` is an odd function, then find the value of `f(5)dot`

Answer 1

`x^(2)f(x)-2f((1)/(x))=g(x) " and " 2f((1)/(x))-4x^(2)f(x)=2x^(2)g((1)/(x))`
or `-3x^(2)f(x)=g(x)+2x^(2)g((1)/(x))`
or ` f(x)=((g(x)+2x^(2)g((1)/(x)))/(3x^(2)))`
Since g(x) is odd `f(-x)= -f(x).`
But given that `f(x)` is even.
` :. f(x)=0`
` :. f(5)=0`