1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
AnnZ [28]
3 years ago
12

How is a corollary related to a theorem.

Mathematics
1 answer:
Serjik [45]3 years ago
6 0
A statement that follows with little or no proof required from an already proven statement. For example, it is a theorem<span> in geometry that the angles opposite two congruent sides of a triangle are also congruent. A </span>corollary<span> to that statement is that an equilateral triangle is also equiangular</span>
You might be interested in
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer d
eduard

The Lagrangian is

L(x_1,\ldots,x_n,\lambda_1,\ldots,\lambda_n)=x_1+\cdots+x_n+\lambda_1({x_1}^2+\cdots+{x_n}^2)+\cdots+\lambda_n({x_1}^2+\cdots+{x_n}^2)

with partial derivatives (set equal to 0)

\dfrac{\partial L}{\partial x_i}=1+2x_i(\lambda_1+\cdots+\lambda_n)=0

\dfrac{\partial L}{\partial\lambda_i}={x_1}^2+\cdots+{x_n}^2-36=0

for each 1\le i\le n.

Let \Lambda be the sum of all the multipliers \lambda_i,

\Lambda=\displaystyle\sum_{k=1}^n\lambda_k=\lambda_1+\cdots+\lambda_n

We notice that

x_i\dfrac{\partial L}{\partial x_i}=x_i+2{x_i}^2\Lambda=0

so that

\displaystyle\sum_{i=1}^nx_i\dfrac{\partial L}{\partial x_i}=\sum_{i=1}^nx_i+2\Lambda\sum_{i=1}^n{x_i}^2=0

We know that \sum\limits_{i=1}^n{x_i}^2=36, so

\displaystyle\sum_{i=1}^nx_i+2\Lambda\sum_{i=1}^n{x_i}^2=0\implies\sum_{i=1}^nx_i=-72\Lambda

Solving the first n equations for x_i gives

1+2\Lambda x_i=0\implies x_i=-\dfrac1{2\Lambda}

and in particular

\displaystyle\sum_{i=1}^nx_i=-\dfrac n{2\Lambda}

It follows that

-\dfrac n{2\Lambda}+72\Lambda=0\implies\Lambda^2=\dfrac n{144}\implies\Lambda=\pm\dfrac{\sqrt n}{12}

which gives us

x_i=-\dfrac1{2\left(\pm\frac{\sqrt n}{12}\right)}=\pm\dfrac6{\sqrt n}

That is, we've found two critical points,

\pm\left(\dfrac6{\sqrt n},\ldots,\dfrac6{\sqrt n}\right)

At the critical point with positive signs, f(x_1,\ldots,x_n) attains a maximum value of

\displaystyle\sum_{i=1}^nx_i=\dfrac{6n}{\sqrt n}=6\sqrt n

and at the other, a minimum value of

\displaystyle\sum_{i=1}^nx_i=-\dfrac{6n}{\sqrt n}=-6\sqrt n

4 0
4 years ago
Which capital letter appears to have parallel line segments
lawyer [7]
The capital letter H since the two lines opposite are parallel
8 0
3 years ago
A triangle, whose sides are 6, 8, and 10, and a circle, whose radius is r, are drawn so that no part of the triangle lies outsid
yulyashka [42]
The radius is half the longest side, 5.
3 0
4 years ago
Read 2 more answers
Is √2/3 a rational or irrational
scoundrel [369]

sqrt(2/3) = 0.816496

since it is a decimal with no repeating numbers it would be irrational

4 0
4 years ago
Read 2 more answers
Please help
Crazy boy [7]

Answer:

157 / 110

Step-by-step explanation:

1.42727........

1000x = 1427.2727.......

10x = 14.2727....... ( - )

___________________

990x = 1413

___________________

x = 1413 / 990

x = 157 / 110

Therefore,

1.42727........ in fraction is 157 / 110.

8 0
3 years ago
Read 2 more answers
Other questions:
  • What is unit form and standard form
    10·2 answers
  • A grocery store sells 6 bottles of water for $4.00 ad 15 bottles of water for $10.00. Is the cost of the water proportional to t
    8·1 answer
  • What is the average of 15, 10, 11, 13, 11?
    10·1 answer
  • Which values of P and Q result in an equation with no solutions?
    10·1 answer
  • A recipie calls for 2 3/4 cups of milk.How many cups of milk,written as a fraction greater than one,are used in the recipie
    8·2 answers
  • The cylinders are similar. The volume of the larger cylinder is 9648 cubic inches. What is the volume of the smaller cylinder?
    12·1 answer
  • Please help me figure out the answer to this question and the processes well!
    11·1 answer
  • Graph the line with the equation y= -2/5x -4<br><br><br>NEED ASAP PLS​
    11·1 answer
  • Rational Number is defined as the number which can be written in a ratio of two integers. An irrational number is a number which
    8·1 answer
  • The graph represents a function.
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!