if i will find the extreme point
f'(x)=0
then
i need to put points and
and if f'(x2)>0 and f'(x1)<0
then its a maximum point
i cant see in your explanation any thing that could tell how to get the number of solution regarding the finding of a maximum point
??
(this function is not a parabola we build a delta >0 in order to deside that there are two solutions)
You're right. The max is located at and so the actual maximum value is
Now clearly if then and since f is continuous on with f approaching negative infinity as then there are x values say where where and so by the intermediate value theorem there exist where giving two solutions to your problem. Similary for and
the intermediate value theorem says that if
there are two points for which f(x1)<0 and f(x2)>0
then there is a x point between then which is a solution
f(x)=0
i found only one such point in the maximum point
there is no another case where f(x1)<0 and f(x2)>0
from where you got the second solution fact??
our function is
f(x)=ln(x)-ax
our extreme point is x=1/a
f''(x)=-1/x^2 if f''(x1)>0 X1 is minimum if f''(x2)<0 X2 is maximum
f(1/a)=ln(1/a)-1
we cant know if its minum or maximum because it depends on the value of
parameter a.
we cant assume that our derivative will get a possitive or negative value
when we input to the second derivative a parameter
so i am confused regarding your solution