question_answer

The smallest natural number c for which the equation \[{{e}^{x}}=c{{x}^{2}}\] has exactly three real and distinct solutions, is

A)  1      

B)                                     2

C)  3    

D)                                     4

Correct Answer: B

Solution :

            \[f(x)\,=\frac{{{e}^{x}}}{{{x}^{2}}}\]             Let \[f(x)=\frac{{{e}^{x}}}{{{x}^{2}}},f'(x)=\frac{(x-2)\,{{e}^{x}}}{{{x}^{3}}}\]             So, from above graph, \[c\,>\,\frac{{{e}^{2}}}{4}\]             So, \[{{c}_{\min }}\,=2\]