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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] ? |
Answer» <html><body><p>`0 lt <a href="https://interviewquestions.tuteehub.com/tag/f-455800" style="font-weight:bold;" target="_blank" title="Click to know more about F">F</a>(x) lt oo` <br/>`-1/2 lt f(x)lt 1/2` <br/>`-1/4 lt f(x) lt1`<br/>`-oo lt f(x) lt 0`</p>Solution :We have ,<br/> `f''(x)-2f(x)+f(x) ge <a href="https://interviewquestions.tuteehub.com/tag/e-444102" style="font-weight:bold;" target="_blank" title="Click to know more about E">E</a>^x " for all " x in [0,1]` <br/> `e^(x)f''(x)-2f''(x)e^(-x)+f(x)e^(-x)f(x)e^(-x)ge 1 " for all " x in [0,1]` <br/> `rArr {e^(-x) f''(x) -e^(-x)f(x)}-{-e^(-x))ge 1 " for all " x in [0,1]` <br/> `rArrd/(dx) f(x)e^)(-x)-f(x)e^(-x) ge 1 " for all " x in [0,1]` <br/> `rArr d/dx{d/dx f(x)e^(-x)} ge 1 " for all " x in [0,1]` <br/> `rArr d^2/dx^2(f(x)d^(-x) ge 1 " for all " x in [0,1]` <br/> `rArrd^2/dx^2( <a href="https://interviewquestions.tuteehub.com/tag/phi-599602" style="font-weight:bold;" target="_blank" title="Click to know more about PHI">PHI</a> (x))ge 1 " for all " x in [0,1] " where " phi (x)=f(x)e^(-x)`<br/>`rArr(x) ` is concave <a href="https://interviewquestions.tuteehub.com/tag/upward-721781" style="font-weight:bold;" target="_blank" title="Click to know more about UPWARD">UPWARD</a> on [0,1]<br/> It is given that f(0)=f(1)=0. Therefore`phi(0)= phi(1)=0` <br/> Therefore<br/> `phi(x)lt 0 " for all" x in (0,1) rArr - oo lt f(x) lt 0 " for all " x in (0,1)`</body></html> | |