Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume that f is differentiable at the points in question. a. The fact that f x (2, 2) = f y (2, 2) = 0 implies that f has a local maximum , local minimum , or saddle point at (2, 2). b. The function f could have a local maximum at ( a , b ) where f y ( a , b ) ≠ 0. c. The function f could have both an absolute maximum and an absolute minimum at two different points that are not critical points. d. The tangent plane is horizontal at a point on a smooth surface corresponding to a critical point.
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume that f is differentiable at the points in question. a. The fact that f x (2, 2) = f y (2, 2) = 0 implies that f has a local maximum , local minimum , or saddle point at (2, 2). b. The function f could have a local maximum at ( a , b ) where f y ( a , b ) ≠ 0. c. The function f could have both an absolute maximum and an absolute minimum at two different points that are not critical points. d. The tangent plane is horizontal at a point on a smooth surface corresponding to a critical point.
Solution Summary: The author explains that the function f has a saddle point at (a,b) and the derivative value is zero.
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume that f is differentiable at the points in question.
a. The fact that fx(2, 2) = fy(2, 2) = 0 implies that f has a local maximum, local minimum, or saddle point at (2, 2).
b. The function f could have a local maximum at (a, b) where
f
y
(
a
,
b
)
≠
0.
c. The function f could have both an absolute maximum and an absolute minimum at two different points that are not critical points.
d. The tangent plane is horizontal at a point on a smooth surface corresponding to a critical point.
Formula Formula A function f(x) attains a local maximum 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 case, f(a) attains a local maximum value f(x) at x=a .
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Chapter 15 Solutions
MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)
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