If the function f(x) is defined by f(x) = a + bx and
`f^(r) = fff` … (respeated r times ) , then `f^(r) (x) ` is equal to

Options

A. `a + b^(r)x`
B. `ar + b^(r) x`
C. ` ar + bx^(r)`
D. `a ((b^(r) -1)/(b -1)) + b^(r)x`

Answer

D

Explanation

` because f(x) = a + bx `
`f{f(x)} = a + b (a + bx)`
` = ab + a + b^(2) x = a (1 +b) + b^(2) x `
`f[f{f(x)}] = f{a(1 + b) +b^(2)x} = a + b {a(1 + b) + b^(2) x}`
` = a (1 + b + b^(2)) + b^(3) x `
`therefore f'(x) = a (1 + b + b^(2)+... + b^(r-1)) + b^(r)x`
` = a ((b^(r) -1)/(b-1)) + b^(r) x`