Isn't it infinity?
How many functions are there such that for every , and and f(0)=0 ? If the number of functions is , then submit your answer as .
Notation: means second derivative of .
The answer is 0.
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1 solution
This is a problem i made from a rather very easy problem into a typical one . Consider a function g(x)= x f ( x ) g'(x)= x ∗ x x f ′ ( x ) − f ( x ) and consider h(x)=xf'(x)-f(x) . h(0)=0 and therefore g'(x)=0 . also h'(x)=x*f''(x) . For x>0 h'(x)>0 . So h(x) is increasing for x>0 . and h(0)=0 . Therefore h(x)>0 for x>0 . Since denominator of g'(x) is positive g'(x) is greater than 0 . therefore g(x) is increasing !!! . but the above data g(1)=-5 and g(2)=-5.5 i.e. g(1)>g(2) . which is not possible as g(x) is increasing . therefore there cannot be such a function .