Use Theorems 8.3 and 8.4, if possible, to find all local minima and maxima of each of the following functions on the giv
Posted: Thu May 12, 2022 9:44 am
Use Theorems 8.3 and 8.4, if possible, to find all local minima
and maxima of each of the following functions on the given
interval. f (x) = ex /x2; [0.5, 3]
Theorem 8.3 (First Derivative Test). Assume that f (x) is
continuous on I = [a, b]. Furthermore, suppose that f (x) is
defined for all x ∈ (a, b), except possibly at x = p. (i) If f (x)
< 0 on (a, p) and f (x) > 0 on (p, b), then f (p) is a local
minimum. (ii) If f (x) > 0 on (a, p) and f (x) < 0 on (p, b),
then f (p) is a local maximum.
Theorem 8.4 (Second Derivative Test). Assume that f is
continuous on [a, b] and f and f are defined on (a, b). Also,
suppose that p ∈ (a, b) is a critical point where f (p) = 0. (i) If
f (p) > 0, then f (p) is a local minimum of f . (ii) If f (p)
< 0, then f (p) is a local maximum of f . (iii) If f (p) = 0,
then this test is inconclusive.
and maxima of each of the following functions on the given
interval. f (x) = ex /x2; [0.5, 3]
Theorem 8.3 (First Derivative Test). Assume that f (x) is
continuous on I = [a, b]. Furthermore, suppose that f (x) is
defined for all x ∈ (a, b), except possibly at x = p. (i) If f (x)
< 0 on (a, p) and f (x) > 0 on (p, b), then f (p) is a local
minimum. (ii) If f (x) > 0 on (a, p) and f (x) < 0 on (p, b),
then f (p) is a local maximum.
Theorem 8.4 (Second Derivative Test). Assume that f is
continuous on [a, b] and f and f are defined on (a, b). Also,
suppose that p ∈ (a, b) is a critical point where f (p) = 0. (i) If
f (p) > 0, then f (p) is a local minimum of f . (ii) If f (p)
< 0, then f (p) is a local maximum of f . (iii) If f (p) = 0,
then this test is inconclusive.