Question #826c5

1 Answer
Jan 30, 2017

Answer:

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)