Critical points and optimization - Math 1A, section 103
1. Find the minimum value of the function f(x) = x2.
2. Find the values of x at which the following functions have a critical point.
(a) f(x) = 2x3 - 3x2 - 36x
(b) g(t) = 1 + t2 + t3 + t4
(c) f(x) = 2 cos(x) + sin2(x)
(d) g(t) = |3t - 4|
(e) h(p) = p-1/p2+4
3. Find all local and absolute maxima and minima of the following functions on the given intervals. Where is the function increasing? Where is it decreasing?
(a) f(x) = x3 - 6x2 + 5, [-3, 5]
(b) f(x) = (x2 - 1)3, [-1, 2]
(c) f(x) = x - ln x, [1/2, 2]
(d) f(t) = t√(4 - t2), [-1, 2]
4. Where is the function f(x) = x3 + x concave up? Concave down?
5. Use the Mean Value theorem to show that |sin(a) - sin(b)| ≤ |a - b| for all a and b.