Question

Show that f(x) = x^3 - e^x has at least one real root using a theorem (calculus)

Answer 1

Using an intermediate value theorem makes sense but I don't know how to use it in this case.

If I substitute -infinity in place of x, the above function is negative. If I substitute positive infinity though, it will also give me a negative number.

Answer 2

Well, all you would need are two numbers, a and b, such that f(a) < 0 and f(b) > 0.

Answer 3

So I just have to show the graph crosses the x-axis. OH! I always thought that you have to substitute in -inf and + inf. So I can substitute in like -1000 and 3

Answer 4

Yeah, you can choose ANY two numbers you wish. f(0) = -1, f(3) = approximately 6.91. Boom, done. Now not sure how fancy shmancy you have to be with any sort of proof statement, but thats all you need.

Answer 5

The choice of x=3 is clever.