Question

I really dont understand how to solve optimization problems at all. Can someone help me?
I have a huge test tomorrow that i cannot afford to fail :/

Answer 1

Take the derivative of a function and set that equal to 0 to get the minimums and maximums

Answer 2

This is because taking the derivative gives the slope of the function, and where the slope is 0 is where the graph changes from either (increasing to decreasing) max, or (decreasing to increasing) minimum

Answer 3

You may have to use multiple equations and substitution rules, but the main principle is setting the derivative equal to 0

Answer 4

Okay but what do you do with the results of finding the min and max?

Answer 5

The intuitive way to understand this rule is to think about what the derivative means. Remember, the derivative is just the slope of the tangent line. It is the instantaneous rate of change, it basically tells you what the function is doing at a given point. So, if the derivative is negative, then the slope of the tangent line is negative, so the function is decreasing. The values are getting smaller. If the derivative is positive, then the slope of the tangent line is positive, so the function is increasing. The values are getting bigger. If you look at the derivative at a point and see that it is negative on the left, and positive on the right, then you know the function was decreasing to the left of the point and increasing to the right of the point. That means you're at a local minimum, that should be very intuitive. Similarly, if the derivative was positive on the left and negative on the right, it should be very intuitive that you're at a local maximum. Since the only way it can go from negative to positive or vice versa is by crossing through zero (or being undefined), it should make good sense that these local extrema occur when the derivative is zero.

Answer 6

say you want to find the minimum of....
y=5x^2 +10
first, take the derivative....
y'=10x
Then set this equal to 0
0=10x
x=0
So we know we have a minimum at x=0
NOTE: You can check this by graphing or by taking the second derivative. If y'' is negative, its a max, if y'' is positive you have found a minimum

Answer 7

in this case....
y''=10, so you have found a minimum

Answer 8

Okay but in a problem like:

Find two numbers whose product is negative 16 and whose sum of squares is a minimum.

How does one approach that?

Answer 9

You would set a variable for each number, let's say \(a\) and \(b\). Then we can say that their product is -16: \(ab=-16\), and we can write the sum of their squares as \(f(a,b)=a^2+b^2\), then differentiate that function, \(f'(a,b)=2a+2b\), set it equal to zero, \(2a+2b=0\), so we know that \(2a=-2b, a=-b\). So we can substitute into the first function and get \(-b^2=-16\), so \(b=4,a=-4\).

Something like that, anyway.

Answer 10

Okay. This is helping me more than you know. I have just one last question though.

Can the method that you've just shown me apply to a question like:

We need to enclose a field with a fence. We have 500 feet of fencing material and a building is on one side of the field and so it won't need any fencing. Determine the dimensions of the field that will enclose the largest area.

Answer 11

Are we supposed to be constructing a rectangle with this fence? If so, we have \(2l+w=500\), \(A=wl\). We can just do algebraic manipulation to see \(w=-2l+500\), so \(A=l(-2l+500)\), so \(A=-2l^2+500l\). Maximize A wrt l by deriving and setting equal to zero: \(-4l+500=0, 4l=500, l=125\). So you want the dimensions to be 125x250.

Answer 12

THANK YOU SO MUCH :')