Question #826c5

1 Answer
Jan 30, 2017

The critical points are

A(sqrt3, sqrt3/6)A(3,36) and B(-sqrt3, -sqrt3/6)B(3,36).

Explanation:

First, we will take the derivative of FF:

(dF)/(dt) = (t^2 + 3 - 2t^2)/(t^2+3)^2 = (-t^2 + 3)/(t^2+3)^2dFdt=t2+32t2(t2+3)2=t2+3(t2+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)