Determine all positive numbers a for which the curve y=a:x intersects the line y=x without calculus The answer is 0<a<e:(1/e) , but how to find it? Is it a system of equations? Which ones? I just need an idea at least, because I'm stuck. If it is impossible without calculus, solve it with it.
aanpalendmw
Answered question
2022-07-19
Determine all positive numbers for which the curve intersects the line without calculus The answer is , but how to find it? Is it a system of equations? Which ones? I just need an idea at least, because I'm stuck. If it is impossible without calculus, solve it with it.
Answer & Explanation
Osvaldo Crosby
Beginner2022-07-20Added 12 answers
We have
This equation is indeed unsolvable without some amount of calculus. Using calculus, however, we may show that is a function that increases from at , reaches its maximum of at , and then decreases as towards With that in mind, it is clear that the curve will intersect the curve if and only if , which is to say that
prkosnognm
Beginner2022-07-21Added 5 answers
This is just a complement to previous answers. As already told by Omnomnomnom
connot be solved in terms of elementary functions. The only explicit solution is given using Lambert function
which has been worked by Euler too; in particular, Euler showed that this equation has a vertical asymptote corresponding to