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If 𝜙(x, y) = 0, show that the determinant
|
fxx + λϕxx
fxy + λϕxy
ϕx
fxy + λϕxy
fyy + λϕyy
ϕy
ϕx
ϕy
|
where 𝜆 is Lagrange’s multiplier, is positive, in case the function attains a maximum.
Let
f
be a differentiable function on
[, ]
and
x [, ].
Show that, if
f (x) 0
and
f (x) 0,
then
f
must have a local maximum at
x.
evaluate limit n tends to infinity [n/(1+n^2) + n/(4+n^2) + n/(9+n^2) +.....+n/2n^2
Show that if n is a natural number and α, β are real numbers with β > 0 then there exists a real function f with derivatives of all orders such that: (i) |f(k)(x)| ≤ β for k ∈ {0, 1, ..., n − 1} and x ∈ (−∞, ∞); (ii) f(k)(0) = 0 for k ∈ {0, 1, ..., n − 1}; (iii) f (n)(0) = α.
Suppose that y = f(x) : (−∞, ∞) → (−∞, ∞) is infinitely differentiable and has a local minimum at 0. Prove that there exists a disc centered on the y axis which lies above the graph of f and touches the graph at the point (0, f(0)).
] Show that if n is a natural number and α, β are real numbers with β > 0 then there exists a real function f with derivatives of all orders such that: (i) |f(k)(x)| ≤ β for k ∈ {0, 1, ..., n − 1} and x ∈ (−∞, ∞); (ii) f(k)(0) = 0 for k ∈ {0, 1, ..., n − 1}; (iii) f (n)(0) = α.
Use POLYA'S FOUR-STEP PROBLEM SOLVING STRATEGY to solve the problems.
- Layla is going to buy 30 multi-vitamin capsules, some P10 and some P18. If she has P340, what is the maximum number of P18 multi-vitamin capsules she can buy? (HINT: Use inequality)
- Yesterday, Ling got confused about which day of the week it was. Whenever they go to long vacation, he forgets. Johnson said its Friday, but Roger said its Saturday. Ling asked what day is it tomorrow? Roger said its Monday, while Johnson said its Tuesday. Ling asked again, what day has it yesterday? Johnson said Wednesday, while Roger said its Thursday. Johnson and Roger have given one correct answer and two wrong answers. What day is it today?
show that the inequalities satisfies for all point x,y∈R
d*(x,y) ≤d(x,y)≤√n d*(x,y)
- Let(a↓n) ↓n€B be any sequence. Show that lim↓n-->♾️ a↓n =Liff for every E>0, there exists some N€N such that n>=N implies a↓n N↓e (L)
