SOLUTION: Find the number which when raised by itself, it will become minimum.

Question 390517: Find the number which when raised by itself, it will become minimum.
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Let . We have to restrict the domain of y to (0, ), otherwise we encounter a densely infinite number of problems, haha.(E.g., , or ). Note also that y > 0 for all x in the domain.
That said, we can take ln of both sides of :
ln y = x lnx.
Differentiating both sides:
<-----(A)
Set , so that the equation (A) becomes lnx = -1, or . Since there is only 1 critical value this is either an absolute min or absolute max. We find out using the 2nd derivative test.
Differentiate (A) again:
.
When x = 1/e, the equation becomes , or . Thus the graph of the function y is concave up at x=1/e, and so there is an absolute minimum there for y.
RELATED QUESTIONS
Find the number which when decreased by three-fourths of itself equals... (answered by checkley79)

Find the number which when decreased by three-fourths of itself equals... (answered by satyareddy22)

What is the number increased by one-half of itself is equal to 21? Find the number which... (answered by ewatrrr)

What number when multiplied by itself is 13 greater than the preceding number when... (answered by Aldorozos)

What number when multiplied by itself is 13 greater than the preceding number when it is... (answered by josgarithmetic)

What number, when multiplied by itself, is 31 greater than the preceding number when it... (answered by ikleyn)

when the digit of a two digit number are reversed it become 5/6 of itself find the number (answered by lwsshak3)

What is the number which when multiplied by itself gives... (answered by ikleyn)

A number becomes 3 times itself when it is increased by 6. What is the number? (answered by Cromlix)