
We have taken for granted that if f'(x) is positive, the function f is _______, and if f'(x) is negative, f is _______. In this section, we prove this rigorously using an important result called the _____ _____ _____. Then we develop a method for "testing" critical points, that is, for determining whether they correspond to _____ _____ or ______
 increasing
 decreasing
 the Mean Value Theorem (MVT)
 local minima or maxima

The MVT says that a secant line between two points (a,f(a)) and (b,f(b)) on a graph is ______ to at least one tangent line in the interval (a,b) (fig. 1 pg 184).
parallel

Since the secant line between (a,f(a)) and (b,f(b)) has the slope ________ and since two lines are parallel if they have the same ______, the MVT is claiming that there exists a point c between a and b such that:
 [f(b)  f(a)]/(b  a)
 slope
 such that:
 f'(c)_{slope of the tangent line} = [f(b)  f(a)]/(b  a)_{slope of the secant line}

State the Mean Value Theorem
 Assume that f is continuous on the closed interval [a,b] and differentiable on (a,b). Then there exists at least one value c in (a,b) such that
 f'(c) = [f(b)  f(a)]/ (b  a)

______ ______ is the special case of the MVT in which f(a) = f(b), in this case, the conclusion is that f'(c)= 0
Rolle's Theorem (4.2)

Pt I Corollary: If f is differentiable and f'(x) = 0 for all x ∈ (a,b), then f is _____ on (a,b). (Explain)
 constant
 In other words, f(x) = C for some constant C

Pt II Corollary: Prove Corollary
 If a1 and b1 are any two distinct points in (a,b), then, by the MVT, there exists c between a1 and b1 such that:
 f(b1)  f(a1) = f'(c)(b1  a1) = 0 (since f'(c) = 0)
 Thus, f(b1) = f(a1). This says that f(x) is constant on (a,b)

According to the behavior, what is happening on the interval (a,b)
Scenario 1: If f(x_{1}) < f(x_{2}) for all x_{1}, x_{2} ∈ (a,b) such that x_{1} < x_{2}.
Scenario 2: If f(x_{1}) > f(x_{2}) for all x_{1}, x_{2} ∈ (a,b) such that x_{1} < x_{2}
When do we say f is monotonic on an interval (a,b)?
 Scenario 1: It is increasing on (a,b)
 Scenario 2: It is decreasing on (a,b)
 We say f is monotonic on (a,b) if it is either increasing or decreasing on (a,b)

State: The Sign of the Derivative Theorem
 Let f be a differentiable function on an open interval (a,b):
 If f'(x) > 0 for x ∈ (a,b), then f is increasing on (a,b)
 If f'(x) < 0 for x ∈ (a,b), then f is decreasing on (a,b)

There is a useful test for determining whether a critical point is a min, a max or neither based on the ____ ____ of the derivative f'(x)
sign change

To explain the term "sign change," suppose that a function g satisfies g(c) = 0. When do we say that g(x) changes from positive to negative at x = c?
If g(x) > 0 to the left of c and g(x) < 0 to the right of c for x within a small open interval around c.
**A sign change from negative to positive is defined similarly. Observe in fig 7 (pg 186) that g(5) = 0 but g(x) does not change sign at x = 5

Now suppose that f'(c) = 0 and that f'(x) changes sign at x = c, say, from + to . Then f is ______ to the left of c and _______ to the right, so f(c) is a ______ _______. Similarly, if f'(x) changes sign from  to +, then f(c) is a ______ _______. (Fig 8A pg 186)
 increasing
 decreasing
 local maximum
 local minimum

Figure 8B illustrates a case where f'(c) = 0 but f'(x) does not change sign. In this case, f'(x) > 0 for all x near but not equal to c, so f is _______ and has neither a _____ ____ or ____ ____ at c. The same analysis holds true when f'(c) ____ ____ ____
 increasing
 local min or a local max
 does not exist

State the First Derivative Test for Critical Points Theorem
 Let c be a critical point of f, then:
 f'(x) changes from + to  at c ⇒ f(c) is a local maximum
f'(x) changes from  to + at c ⇒ f(c) is a local minimum

To carry out the First Derivative Test, we make a useful observation: f'(x) can change sign at a _____ _____, but it cannot change sign on the interval between ____ _____ ______ _____ (this can be proved even if f' is not assumed to be continuous). So we can determine the sign of f'(x) on an interval between ______ _____ _____ by evaluating f'(x) at an any test point x_{0} inside the interval. The sign of f'(x_{0}) is the sign of f'(x) on the entire interval. (Fig 8A and 8B pg 186)
 critical point
 two consecutive critical points
 consecutive critical points

