4.3 The Mean Value Theorem and Montonicity

  1. 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
  2. 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).
  3. 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
  4. 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)
  5. ______ ______ 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)
  6. 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
  7. 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)
  8. According to the behavior, what is happening on the interval (a,b)
    Scenario 1: If f(x1) < f(x2) for all x1, x2 ∈ (a,b) such that x1 < x2.
    Scenario 2: If f(x1) > f(x2) for all x1, x2  ∈ (a,b) such that x1 < x2
    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)
  9. 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)
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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 x0 inside the interval. The sign of f'(x0) 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
Card Set
4.3 The Mean Value Theorem and Montonicity
Section 4.3