How does the domain change when finding the extremum of a function with and without constraints

if we want to find the maximum or minimum of a function, there are some methods, like see where the derivate is equal to zero and such.

Now, if we have a constraint, for example x> 4

This obviously changes the domain of the possible points where we can find the extremum.

But particularly, we can check at x  = 4 and see it there is a maximum.

if the constraint is x > 4, then then we may found a upper/lower bound.

if the constraint is x ≥ 4, then we may find a maximum/minimum.

Now, when we work with more complex situations, like find the maximum value of f(x) while we have the constraint g(x) = x^4 + x^3 + x^2 = 5

Then the domain will be equal to the intersection of the domains of both functions:

This means, the only possible values that we can input in f(x) to search the extremes, are the values of x such that x^4 + x^3 + x^2 = 5, so this relation defines our new domain (remember that f(x) also may have problems in some values of x, this is why we take the intersection of the domains of f(x) and g(x))

This means that constraints can limit a lot the domain of our functions.

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step-by-step explanation:

the total is 23.97. 0.25% is off so you have to subtracting 25% from 100% and you will get 75% if you convert it to decimal you will get 0.75. you have to find the original because the formula is % times original = total.so you have the total and the % so you just have to plug it in.

step-by-step explanation: