Use the equation ƒ(x) = ƒ(a) + ƒ′(a)(x - a) + ƒ′′(c2) /2 (x - a)2 to establish the following test. Let ƒ have continuous first and second derivatives and suppose that ƒ′(a) = 0. Then a. ƒ has a local maximum at a if ƒ′′<= 0 throughout an interval whose interior contains a; b. ƒ has a local minimum at a if ƒ′'>= 0 throughout an interval whose interior contains a.
Use the equation ƒ(x) = ƒ(a) + ƒ′(a)(x - a) + ƒ′′(c2) /2 (x - a)2 to establish the following test. Let ƒ have continuous first and second derivatives and suppose that ƒ′(a) = 0. Then a. ƒ has a local maximum at a if ƒ′′<= 0 throughout an interval whose interior contains a; b. ƒ has a local minimum at a if ƒ′'>= 0 throughout an interval whose interior contains a.
Use the equation ƒ(x) = ƒ(a) + ƒ′(a)(x - a) + ƒ′′(c2) /2 (x - a)2 to establish the following test. Let ƒ have continuous first and second derivatives and suppose that ƒ′(a) = 0. Then a. ƒ has a local maximum at a if ƒ′′<= 0 throughout an interval whose interior contains a; b. ƒ has a local minimum at a if ƒ′'>= 0 throughout an interval whose interior contains a.
Use the equation ƒ(x) = ƒ(a) + ƒ′(a)(x - a) + ƒ′′(c2) /2 (x - a)2 to establish the following test. Let ƒ have continuous first and second derivatives and suppose that ƒ′(a) = 0. Then a. ƒ has a local maximum at a if ƒ′′<= 0 throughout an interval whose interior contains a; b. ƒ has a local minimum at a if ƒ′'>= 0 throughout an interval whose interior contains a.
Formula Formula A function f ( x ) is also said to have attained a local minimum at x = a , if there exists a neighborhood ( a − δ , a + δ ) of a such that, f ( x ) > f ( a ) , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a f ( x ) − f ( a ) > 0 , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a In such a case f ( a ) is called the local minimum value of f ( x ) at x = a .
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