Show that the function has exactly one zero in the given interval.

Question

Show that the function has exactly one zero in the given interval.

g(t) = √t + √(1+t) - 4, (0, ∞)

Answer 1

Hey Alex,

I'm not sure exactly how much depth you're going to need to completely prove this one, but here's a few things to get you started.

First of, we know that our function is continuous on the given interval. At the start of our interval (x = 0), we have a starting output of g(0) = -3.

From there, we can take the derivative of g(t). I won't do that here as it will be a bit cumbersome to type, but your resulting function, g'(t) will always be positive on the open interval given. What that means is that g(t) is strictly increasing. In layman's terms, it can only go up, and it will never come back down.

So since it's continuous, you can pick some larger x value that gives you a positive output. Since we start at a negative output, the Intermediate Value Theorem would then guarantee us at least one zero, but since we've shown that g(t) is strictly increasing, there can't be a second.

Hope that gets you started in the right direction!