File: Mathlib/Analysis/NormedSpace/Extr.lean

full_name: IsMaxFilter.norm_add_sameRay
code: /-- If `f : α → E` is a function such that `norm ∘ f` has a maximum along a filter `l` at a point
`c` and `y` is a vector on the same ray as `f c`, then the function `fun x => ‖f x + y‖` has
a maximum along `l` at `c`. -/
theorem IsMaxFilter.norm_add_sameRay (h : IsMaxFilter (norm ∘ f) l c) (hy : SameRay ℝ (f c) y) :
    IsMaxFilter (fun x => ‖f x + y‖) l c 
goal: IsMaxFilter (fun x => ‖f x + y‖) l c

full_name: IsMaxFilter.norm_add_self
code: /-- If `f : α → E` is a function such that `norm ∘ f` has a maximum along a filter `l` at a point
`c`, then the function `fun x => ‖f x + f c‖` has a maximum along `l` at `c`. -/
theorem IsMaxFilter.norm_add_self (h : IsMaxFilter (norm ∘ f) l c) :
    IsMaxFilter (fun x => ‖f x + f c‖) l c 
goal: IsMaxFilter (fun x => ‖f x + f c‖) l c

full_name: IsMaxOn.norm_add_sameRay
code: /-- If `f : α → E` is a function such that `norm ∘ f` has a maximum on a set `s` at a point `c` and
`y` is a vector on the same ray as `f c`, then the function `fun x => ‖f x + y‖` has a maximum
on `s` at `c`. -/
theorem IsMaxOn.norm_add_sameRay (h : IsMaxOn (norm ∘ f) s c) (hy : SameRay ℝ (f c) y) :
    IsMaxOn (fun x => ‖f x + y‖) s c 
goal: IsMaxOn (fun x => ‖f x + y‖) s c

full_name: IsMaxOn.norm_add_self
code: /-- If `f : α → E` is a function such that `norm ∘ f` has a maximum on a set `s` at a point `c`,
then the function `fun x => ‖f x + f c‖` has a maximum on `s` at `c`. -/
theorem IsMaxOn.norm_add_self (h : IsMaxOn (norm ∘ f) s c) : IsMaxOn (fun x => ‖f x + f c‖) s c 
goal: IsMaxOn (fun x => ‖f x + f c‖) s c

full_name: IsLocalMaxOn.norm_add_sameRay
code: /-- If `f : α → E` is a function such that `norm ∘ f` has a local maximum on a set `s` at a point
`c` and `y` is a vector on the same ray as `f c`, then the function `fun x => ‖f x + y‖` has a local
maximum on `s` at `c`. -/
theorem IsLocalMaxOn.norm_add_sameRay (h : IsLocalMaxOn (norm ∘ f) s c) (hy : SameRay ℝ (f c) y) :
    IsLocalMaxOn (fun x => ‖f x + y‖) s c 
goal: IsLocalMaxOn (fun x => ‖f x + y‖) s c

full_name: IsLocalMaxOn.norm_add_self
code: /-- If `f : α → E` is a function such that `norm ∘ f` has a local maximum on a set `s` at a point
`c`, then the function `fun x => ‖f x + f c‖` has a local maximum on `s` at `c`. -/
theorem IsLocalMaxOn.norm_add_self (h : IsLocalMaxOn (norm ∘ f) s c) :
    IsLocalMaxOn (fun x => ‖f x + f c‖) s c 
goal: IsLocalMaxOn (fun x => ‖f x + f c‖) s c

full_name: IsLocalMax.norm_add_sameRay
code: /-- If `f : α → E` is a function such that `norm ∘ f` has a local maximum at a point `c` and `y` is
a vector on the same ray as `f c`, then the function `fun x => ‖f x + y‖` has a local maximum
at `c`. -/
theorem IsLocalMax.norm_add_sameRay (h : IsLocalMax (norm ∘ f) c) (hy : SameRay ℝ (f c) y) :
    IsLocalMax (fun x => ‖f x + y‖) c 
goal: IsLocalMax (fun x => ‖f x + y‖) c

full_name: IsLocalMax.norm_add_self
code: /-- If `f : α → E` is a function such that `norm ∘ f` has a local maximum at a point `c`, then the
function `fun x => ‖f x + f c‖` has a local maximum at `c`. -/
theorem IsLocalMax.norm_add_self (h : IsLocalMax (norm ∘ f) c) :
    IsLocalMax (fun x => ‖f x + f c‖) c 
goal: IsLocalMax (fun x => ‖f x + f c‖) c