Suppose `f(x)=e^(a x)+e^(b x)` , where `a!=b ,` and that `f^(prime_ prime)(x)-2f^(prime)(x)-15f(x)=0` for all `xdot` Then the value of `(|a b|)/3` is ___

Options

A. 25
B. 9
C. -15
D. -9

Explanation

`(a^(2)-2a-15)e^(ax)+(b^(2)-2b-15)e^(bx)=0`
`"or "(a^(2)-2a-15)=0 and b^(2)-2b-15=0`
`"or "(a-5)(a+3)=0 and (b-5)(b+3)=0`
`i.e., a=5 or -3 and b=5 or -3`
`therefore" "aneb`
Hence, `a=5 and b=-3 or a =-3 and b=5`
`"or "ab=-15`