True or False (i) Let f : R² —>R be a function. If both df/dx and df/dy exist at ā∈R², then any directional derivative of f also exists at ā (ii) Let D be a closed and bounded region in R² and let f be a C¹ function over D. Suppose that f attains its minimum over D at a point ā ∈ D, then ▽f (ā) = 0. (iii) Let A be a subset of R². If A is not open, then A is closed. (iv) Let F be a C² vector field on R³. Then div(curl F) must be the zero vector field. (v) Let f be a C² function on Rⁿ and let ā be a critical point of f. If Hess f(ā) has both positive and negative eigenvalues, then ā cannot be a local minimum or maximum of f.
True or False (i) Let f : R² —>R be a function. If both df/dx and df/dy exist at ā∈R², then any directional derivative of f also exists at ā (ii) Let D be a closed and bounded region in R² and let f be a C¹ function over D. Suppose that f attains its minimum over D at a point ā ∈ D, then ▽f (ā) = 0. (iii) Let A be a subset of R². If A is not open, then A is closed. (iv) Let F be a C² vector field on R³. Then div(curl F) must be the zero vector field. (v) Let f be a C² function on Rⁿ and let ā be a critical point of f. If Hess f(ā) has both positive and negative eigenvalues, then ā cannot be a local minimum or maximum of f.
True or False (i) Let f : R² —>R be a function. If both df/dx and df/dy exist at ā∈R², then any directional derivative of f also exists at ā (ii) Let D be a closed and bounded region in R² and let f be a C¹ function over D. Suppose that f attains its minimum over D at a point ā ∈ D, then ▽f (ā) = 0. (iii) Let A be a subset of R². If A is not open, then A is closed. (iv) Let F be a C² vector field on R³. Then div(curl F) must be the zero vector field. (v) Let f be a C² function on Rⁿ and let ā be a critical point of f. If Hess f(ā) has both positive and negative eigenvalues, then ā cannot be a local minimum or maximum of f.
True or False
(i) Let f : R² —>R be a function. If both df/dx and df/dy exist at ā∈R², then any directional derivative of f also exists at ā
(ii) Let D be a closed and bounded region in R² and let f be a C¹ function over D. Suppose that f attains its minimum over D at a point ā ∈ D, then ▽f (ā) = 0.
(iii) Let A be a subset of R². If A is not open, then A is closed.
(iv) Let F be a C² vector field on R³. Then div(curl F) must be the zero vector field.
(v) Let f be a C² function on Rⁿ and let ā be a critical point of f. If Hess f(ā) has both positive and negative eigenvalues, then ā cannot be a local minimum or maximum of f.
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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