Question

Question #826c5

Answer 1

The critical points are

`A(sqrt3, sqrt3/6)` and `B(-sqrt3, -sqrt3/6)`.

Explanation:

First, we will take the derivative of `F`:

`(dF)/(dt) = (t^2 + 3 - 2t^2)/(t^2+3)^2 = (-t^2 + 3)/(t^2+3)^2`

We used the quotient rule: `(a/b)' = (a'b - ab')/b^2`.

Now, we must find the values where the derivative is either zero, or undefined. Those will be the critical points. Since the denominator is always positive, it can never be zero, so `F'` is always defined. However, it can be zero, when the numerator is zero:

`F'(t) = 0 => t = +-sqrt3`.

Therefore, the critical points are

`A(sqrt3, sqrt3/6)` and `B(-sqrt3, -sqrt3/6)`.

Extra note:

If we so wanted, we could then find where `F'` is positive or negative to determine if `F` is increasing or decreasing. (and since it's differentiable, we know that it's continuous, so we are sure about that last statement)