# Equilibrium point

• Jan 26th 2010, 10:08 AM
Len
Equilibrium point
Let f(y) be a continuous function:

Suppose that f(-10)>0 and f(10)<0, show that there is a equilibrium point for dy/dt=f(y) between y=-10 and y=10.

Just realized I posted this is in the wrong section, should be in Differential Equations, sorry.
• Jan 26th 2010, 12:07 PM
Calculus26
Recall an equillibrium solution occurs where f(y) = 0 for dy/dt = f(y)

By the Intermediate Value Theorem if f(10) < 0 and f(-10) > 0

Then f has a zero between -10 and 10
• Jan 28th 2010, 08:54 AM
Len
Quote:

Originally Posted by Calculus26
Recall an equillibrium solution occurs where f(y) = 0 for dy/dt = f(y)

By the Intermediate Value Theorem if f(10) < 0 and f(-10) > 0

Then f has a zero between -10 and 10

Thanks, I didn't even think to apply that.

The intermediate value theorem states the following: If the function y = f(x) is continuous on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u.