Calculus: Early Transcendentals, Chapter 4, 4.1, Section 4.1, Problem 55

Given: f(t)=tsqrt(4-t^2),[-1,2].
Find the critical number by setting the first derivative equal to zero and solving for the t values. Find the derivative using the product rule.
f'(t)=t[-t/(4-t^2)^(1/2)]+(4-t^2)^(1/2)=0
t^2/(4-t^2)^(1/2)=(4-t^2)^(1/2)
t^2=4-t^2
2t^2=4
t^2=2
t=+-sqrt(2)
The critical numbers are x=+-sqrt(2).
Substitute the critical numbers and the endpoints of the interval [-1,2] into the original f(t) function. Do NOT substitute in the t=-sqrt(2) because it is not in the interval [-1,2].
f(-1)=-sqrt(3)
f(sqrt(2))=2
f(2)=0
Examine the f(x) values to determine the absolute maximum and absolute minimum.
The absolute maximum occurs at the point (sqrt(2),2) .
The absolute minimum occurs at the point (-1,-sqrt(3)),