Let `f,g` and `h` be real-valued functions defined on the interval `[0,1]` by `f(x)=e^(x^2)+e^(-x^2)` , `g(x)=x e^(x^2)+e^(-x^2)` and `h(x)=x^2 e^(x^2)+e^(-x^2)`. if `a,b` and `c` denote respectively, the absolute maximum of `f,g` and `h` on `[0,1]` then

Options

A. `a=b and c ne b`
B. `a=c and ane b `
C. `a ne b and c ne b `
D. a=b=c

Answer

D

Explanation

For any ` x in [0,1]` we have
`x^2 le x le 1`
` rArr " "x^2 e^(x^2) le x e^(x^2)le e^(x^2)`
`rArr e^(-x^2)+x^(2)e^(x^2)+e^(-x^2)+x e^(x^2) le e^(-x^2 )+e^(x^2)`
`rArr h(x) le g(x) le f(x)`
Now , `f(x)=e^(x^2)+e^(-x^2)`
`rArr f(x)=2x(ex^2=e^(-x^2))gt 0 " for all "x in (0,1]`
`rArr f(x)` is increasing on (0,1]
`rArr f(1)` is the maximum vlaue of f(x) on [0,1]
`rArr a=e+e^(-1)`
Also `f(1)=g(1)=h(1)=e+e^(-1)`
`therefore a=b=c=e+e^(-1)`