Critical points and optimization find the minimum value of

Question

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.