Discussion Forum

Let f:[0,1]→ R (the set of all real numbers) be a function .Suppose  the function f is twice differentiable

 f(0)=f(1) =0 and satisfies

 $f '' (x)-2 f'(x)+ f(x) \geq e^{x}, x \in [0,1]$.

 If the function  e^-x f(x)  assumes its minimum in the interval [0,1] a  $x=\frac{1}{4}$, which of the following is true ?

Answer:

Explanation:

 Concept involved It is based on the concept of converting into total differential equation (i.e, completing the equation into differential ). So , as to check the function to be increasing or decreasing

 Here,  $\phi'(x)<0,x\in (0,\frac{1}{4})$

 and   $\phi'(x)>0,x\in (\frac{1}{4},1)$

  $\Rightarrow $    $e^{-x}f'(x) -e^{-x}f(x)<0, x\in (0,\frac{1}{4})$

$\Rightarrow$     $ f'(x) <f(x) , 0 <x<\frac{1}{4}$