Let f(x) and g(x) be defined and differntiable for all `x ge x_0 and f(x_0)=g(x_0) f(x) ge (x) for x gt x_0` then

Options

A. `f(x)lt g(x) n x gt x_0`
B. `f(x)=g(x) x=x_0`
C. `f(x) gt g (x) ,x le x_0`j
D. none of these

Answer

C

Explanation

Let `h(x)-f (x)-g (x) for x gt g(x) "for all" x le x_0` Since f(x) and f(x) are differentiable for all `x ge x_0` Therefore so is h(x)
Now `h(x)=f(x)-g (x) " for all " x ge x_0`
`rArr h(x) =f' (x)-g'(x)`
`rArr h'(x) gt 0 "for all " x gt x_0 " "[because f(x) gt g (x) for all x lt x_0]`
`rArr` h(x) is an increasing function for all `x lt x_0`
`rArr h(x)` is an increasing function for all `xlt x_0`
`rArr h(x) h(x_0) " for all" x gt x_0`
`rArr h(x) gt 0 for all x gt x_0 " "[ because h(x_0)=f(x_0)-g(x_0)-g(x_0)=0]`
`h(x) gt 0 "for all" x lt x_0`