# GRE Subject Test: Math : Lagrange's Theorem

## Example Questions

### Example Question #321 : Series In Calculus

Let be the fifth-degree Taylor polynomial approximation for , centered at .

What is the Lagrange error of the polynomial approximation to ?      Explanation:

The fifth degree Taylor polynomial approximating centered at is: The Lagrange error is the absolute value of the next term in the sequence, which is equal to .

We need only evaluate this at and thus we obtain ### Example Question #1 : Alternating Series With Error Bound

Which of the following series does not converge?      Explanation:

We can show that the series diverges using the ratio test.   will dominate over since it's a higher order term. Clearly, L will not be less than, which is necessary for absolute convergence.

Alternatively, it's clear that is much greater than , and thus having in the numerator will make the series diverge by the limit test (since the terms clearly don't converge to zero).

The other series will converge by alternating series test, ratio test, geometric series, and comparison tests.

### Example Question #1 : Applications Of Partial Derivatives

Find the minimum and maximum of , subject to the constraint . is a maximum is a minimum is a maximum is a minimum is a maximum is a minimum is a maximum is a minimum

There are no maximums or minimums is a maximum is a minimum

Explanation:

First we need to set up our system of equations.   Now lets plug in these constraints.   Now we solve for  If   If   Now lets plug in these values of , and into the original equation.  We can conclude from this that is a maximum, and is a minimum.

### Example Question #1 : Lagrange Multipliers

Find the absolute minimum value of the function subject to the constraint .      Explanation:

Let To find the absolute minimum value, we must solve the system of equations given by .

So this system of equations is , , .

Taking partial derivatives and substituting as indicated, this becomes .

From the left equation, we see either or . If , then substituting this into the other equations, we can solve for , and get , , giving two extreme candidate points at .

On the other hand, if instead , this forces from the 2nd equation, and from the 3rd equation. This gives us two more extreme candidate points; .

Taking all four of our found points, and plugging them back into , we have .

Hence the absolute minimum value is .

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