Let
and
be functions of two variables with continuous partial derivatives at every point of some open set containing the smooth curve
Suppose that
when restricted to points on the curve
has a local extremum at the point
and that
Then there is a number
called a
Lagrange multiplier , for which
Proof
Assume that a constrained extremum occurs at the point
Furthermore, we assume that the equation
can be smoothly parameterized as
where
s is an arc length parameter with reference point
at
Therefore, the quantity
has a relative maximum or relative minimum at
and this implies that
at that point. From the chain rule,
where the derivatives are all evaluated at
However, the first factor in the dot product is the gradient of
and the second factor is the unit tangent vector
to the constraint curve. Since the point
corresponds to
it follows from this equation that
which implies that the gradient is either
or is normal to the constraint curve at a constrained relative extremum. However, the constraint curve
is a level curve for the function
so that if
then
is normal to this curve at
It follows, then, that there is some scalar
such that
□
To apply
[link] to an optimization problem similar to that for the golf ball manufacturer, we need a problem-solving strategy.
Problem-solving strategy: steps for using lagrange multipliers
Determine the objective function
and the constraint function
Does the optimization problem involve maximizing or minimizing the objective function?
Set up a system of equations using the following template:
Solve for
and
The largest of the values of
at the solutions found in step
maximizes
the smallest of those values minimizes
Using lagrange multipliers
Use the method of Lagrange multipliers to find the minimum value of
subject to the constraint
Let’s follow the problem-solving strategy:
The optimization function is
To determine the constraint function, we must first subtract
from both sides of the constraint. This gives
The constraint function is equal to the left-hand side, so
The problem asks us to solve for the minimum value of
subject to the constraint (see the following graph).
We then must calculate the gradients of both
f and
g :
The equation
becomes
which can be rewritten as
Next, we set the coefficients of
equal to each other:
The equation
becomes
Therefore, the system of equations that needs to be solved is
This is a linear system of three equations in three variables. We start by solving the second equation for
and substituting it into the first equation. This gives
so substituting this into the first equation gives
Solving this equation for
gives
We then substitute this into the third equation:
Since
this gives
Next, we substitute
into
gives
To ensure this corresponds to a minimum value on the constraint function, let’s try some other values, such as the intercepts of
Which are
and
We get
and
so it appears
has a minimum at
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miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
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Samuel
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hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities.
before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue..
sometimes due to these physical problems..the diagnosis may be misdiagnosed.
treatment for autism.
well it depends on the severity.
since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision.
play therapy.
speech therapy.
Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game.
string and beads game.
the more you interact with the child the more effective
Jharna
results you'll get..
please consult a therapist to know what suits best on your child.
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Jharna
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