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Learn about Local Extrema, Rolle’s Theorem, Mean Value Theorem, Fermat’s Theorem, and examples to grasp these concepts. Guidelines for finding local extrema, critical points, applications, and cautionary notes are covered.
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Local Extrema & Mean Value Theorem • Local Extrema • Rolle’s theorem: What goes up must come down • Mean value theorem: Average velocity must be attained • Some applications: Inequalities, Roots of Polynomials
Example • Let f(x) = (x-1)2(x+2), -2 x 3 • Use the graph of f(x) to find all local extrema • Find the global extrema
Example • Consider f(x) = |x2-4| for –2.5 x < 3 • Find all local and global extrema
Fermat’s Theorem • Theorem: If f has a local extremum at an interior point c and f (c) exists, then f (c) = 0. • Proof Case 1: Local maximum at interior point c • Then derivative must go from 0 to 0 around c • Proof Case 2: Local minimum at interior point c • Then derivative must go from 0 to 0 around c
Cautionary notes • f (c) = 0 need not imply local extrema • Function need not be differentiable at a local extremum (e.g., earlier example |x2-4|) • Local extrema may occur at endpoints
Summary: Guidelines for finding local extrema: • Don’t assume f (c) = 0 gives you a local extrema (such points are just candidates) • Check points where derivative not defined • Check endpoints of the domain • These are the three candidates for local extrema • Critical points: points where f (c)=0 or where derivative not defined
What goes up must come down • Rolle’s Theorem: Suppose that f is differentiable on (a,b) and continuous on [a,b]. If f(a) = f(b) = 0 then there must be a point c in (a,b) where f (c) = 0.
Proof of Rolle’s theorem • If f = 0 everywhere it’s easy • Assume that f > 0 somewhere (case f< 0 somewhere similar) • Know that f must attain a maximum value at some point which must be a critical point as it can’t be an endpoint (because of assumption that f > 0 somewhere). • The derivative vanishes at this critical point where maximum is attained.
Need all hypotheses • Suppose f(x) = exp(-|x|). • f(-2) = f(2) • Show that there is no number c in (-2,2) so that f (c)=0 • Why doesn’t this contradict Rolle’s theorem
Examples • f(x) = sin x on [0, 2] • exp(-x2) on [-1,1] • Any even continuous function on [-a,a] that is differentiable on (-a,a)
Average Velocity Must be Attained • Mean Value Theorem: Let f be differentiable on (a,b) and continuous on [a,b]. Then there must be a point c in (a,b) where
Idea in Mean Value Proof • Says must be a point where slope of tangent line equals slope of secant lines joining endpoints of graph. • A point on graph furthest from secant line works:
Consequences • f is increasing on [a,b] if f (x) > 0 for all x in (a,b) • f is decreasing if f (x) < 0 for all x in (a,b) • f is constant if f (x)=0 for all x in (a,b) • If f (x) = g(x) in (a,b) then f and g differ by a constant k in [a,b]
Examples • Show that f(x) = x3+2 satisfies the hypotheses of the Mean Value Theorem in [0,2] and find all values c in this interval whose existence is guaranteed by the theorem. • Suppose that f(x) = x2 – x –2, x in [-1,2]. Use the mean value theorem to show that there exists a point c in [-1,2] with a horizontal tangent. Find c.
Existence/non-existence of roots • x3 + 4x - 1 = 0 must have fewer than two solutions • The equation 6x5 - 4x + 1 = 0 has at least one solution in the interval (0,1)
