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Let f:[0,1] rarr R be a function . Suppose the fuction f is twice differentiable with f(0) =f(1)=0and satisfies f(x)-2f(x)+f(x) ge e^x " for all " x in [0,1].Which of the following is true for x in (0,1] ? |
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Answer» `0 lt F(x) lt oo` `f''(x)-2f(x)+f(x) ge E^x " for all " x in [0,1]` `e^(x)f''(x)-2f''(x)e^(-x)+f(x)e^(-x)f(x)e^(-x)ge 1 " for all " x in [0,1]` `rArr {e^(-x) f''(x) -e^(-x)f(x)}-{-e^(-x))ge 1 " for all " x in [0,1]` `rArrd/(dx) f(x)e^)(-x)-f(x)e^(-x) ge 1 " for all " x in [0,1]` `rArr d/dx{d/dx f(x)e^(-x)} ge 1 " for all " x in [0,1]` `rArr d^2/dx^2(f(x)d^(-x) ge 1 " for all " x in [0,1]` `rArrd^2/dx^2( PHI (x))ge 1 " for all " x in [0,1] " where " phi (x)=f(x)e^(-x)` `rArr(x) ` is concave UPWARD on [0,1] It is given that f(0)=f(1)=0. Therefore`phi(0)= phi(1)=0` Therefore `phi(x)lt 0 " for all" x in (0,1) rArr - oo lt f(x) lt 0 " for all " x in (0,1)` |
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