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1) You may use Lagrange multipliers as one fine method to find the critical points of a constrained multivariable function.
The algorithm you need to follow when using this method is to create another multivariable function, based on original function, called Lagrange function using a Lagrange multiplier, which is a new variable lambda.
If partial derivatives of Lagrange function cancel then the Lagrange function has a stationary point at (x,y,lambda) and the original function f(x,y) is maximum.
2) You may find the maximum or minimum of an unconstrained multivariable function using the zero subgradient. Hence, you need to find the gradient of the function, grad(f) (partial derivatives of function f) and cancel it. The values that cancel gradient denote the stationary points of original function.
3) You may also use the closed interval method to find maxima and minima of a function of a single variable, defined over a closed interval [a,b].
You need to verify if the behaviour of the function is continuous over the interval [a,b]. If the function proves its continuity, then you may differentiate it and then find the critical numbers.
Substituting these critical numbers for x you'll find the critical values of function.
You also need to evaluate the values of the function for the endpoints of interval, a and b.
The final step is to decide the absolute minima or maxima by comparing the endpoints values of function to the critical values. The smallest value denotes an absolute minimum and the largest value denotes an absolute maximum.
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