Let `f:R rarr R, y=f(x), f(0)=0, f'(x) gt0 and f''(x)gt0`. Three point `A(alpha, f(alpha)), B(beta,f(beta)), C(gamma, f(gamma)) on y=f(x)` such that `0lt alpha lt beta lt gamma.`
Which of the following is false ?

Options

A. `alphaf(beta) gt beta(f(alpha))`
B. `alphaf(beta)lt beta f(alpha)`
C. `gamma f(beta)lt beta(f(gamma))`
D. `gamma (f(alpha))lt alpha f(gamma)`

Answer

B

Explanation

Consider `g(x)=(f(x))/(x)`
`g'(x)=(xf'(x)-f(x))/(x^(2))=(f'(x))/(x)(x-(f(x))/(f'(x)))`
`as (f'(x))/(x^(2))gt0`
Let `h(x)=x-(f(x))/(f'(x))`
`therefore" "h'(x)=(f'(x)^(2)-f(x)f''(x))/(f'(x)^(2))`
`=(f(x)f''(x))/(f'(x)^(2))gt0`
`therefore x-(f(x))/(f'(x))` is increasing function.

`x-(f(x))/(f'(x))gt0-(f(0))/(f'(0))=0`
`therefore" "g'(x)gt0`.
`therefore" "(f(gamma))/(gamma)gt(f(beta))/(beta)lt(f(alpha))/(alpha)`
`therefore" "B is false