Consider the function g ( x , y ) = x × y in the domain − 5 ≤ x ≤ 1 0 and − 5 ≤ y ≤ 1 0 . What is the value of
∣ min g ( x , y ) ∣ ?
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oh man!!! it is quite easy but i misread the question. I thought it asks min of abs(g(x,y))=0!!! wow
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Ohh, I can understand. Even I make such mistakes sometimes :)
I did the same mistake.
This property which I mentioned is valid only when the domains of x and y contain negative values in them.
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same mistake....easy but tricky! you tempt to write zero!!
i too solved it as 0..:(
I DID THE SAME WAY
I put 0
Note that in order to minimize g ( x , y ) we wish to create the largest possible negative value. This is obtained when we maximize one variable in the positive range and make the other variable as negative as possible. Thus, the minimum value of g ( x , y ) is − 5 0 and the absolute value of that is 5 0 .
Minimum value will be obtained when we have -|maximum magnitude| which is only possible when we have pairing of -5 and +10 and hence g ( x , y ) = − 5 × 1 0 which is -50 but we have to take |-50| which is 50
min A is the minimum value from the set A .
Now, g ( x , y ) is the range of the function. The range of the function includes elements like { − 5 0 , − 4 5 , 1 , … } . The minimum value here is − 5 0 . The absolute value of ∣ − 5 0 ∣ is 5 0 .
Therefore the answer is 5 0
Draw the 2 number lines of the respective domains , take -5 from one extreme from one number line and 10 [for the other domain] from another extreme , multiply and take mod. is 50.
Since − 5 ≤ x ≤ 1 0 and − 5 ≤ y ≤ 1 0 , we have that the range for x and y includes all the negative reals greater than or equal to − 5 and all the non-negative reals less than or equal to 1 0 . Hence we have that in order to minimise the product, we take the negative lower bound of the domain of one and multiply it by the positive upper bound of the other variable. This gives us a product of − 5 × 1 0 = − 5 0 .
Taking the absolute value yields: ∣ min g ( x , y ) ∣ = ∣ − 5 0 ∣ = 5 0 .
The minimum of this function can be achieved when one variable is at its minimum, and the other is at its maximum, or ( x , y ) = ( − 5 , 1 0 ) Therefore, ∣ min g ( x , y ) ∣ = ∣ − 5 0 ∣ = 5 0
At first we have to find the min value of g(x, y) .
The domain exists in negative range too. So, for the minimum value, we have to find the highest negative value of g(x, y).
It can only be possible if we consider one value positive and other value negative , Then the multiplication will be negative.
So, The highest negative value in the domain is (-)5 . The positive value also should be highest, because the the more the value will be higher, the more the value will be small.
The highest positive value in the domain is 10 .
So, the min g(x, y) = - 5 X 10 = - 50 .
We have to give the answer in modulus form.
So, |min g(x, y)| = |-50| = 50
So, the answer is 50 .
as domain of x and y is same min g(x,y)=min(x) x max(y) or min(y) x max(x) so min g(x,y) = -50 so |min g(x,y)| = 50
−5 ≤ x ≤10 and −5 ≤ y ≤ 10 <=> -50 ≤ x \times y ≤ 100 <=> 50 ≤ |min g(x,y)| ≤ 100. The result is 50
Here, the function g ( x , y ) = x × y . We get the minimum value of the function when the values of x and y are the lowest and highest values of the domain or vice versa. Here, we get min value of g(x,y) in g(-5,10)=g(10,-5)=(-50)
Now, ∣ min g ( x , y ) ∣ = ∣ − 5 0 ∣ = 5 0
Are you sure about the statement "We get minimum value ... when they are the lowest and highest values?"
What about the function f ( x , y ) = x × y in the domain 5 ≤ x ≤ 1 0 , 5 ≤ y ≤ 1 0 ? Is the (absolute value of the) minimum going to be 50?
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No, since here the domain has only positive real values and no negative values, so in this case we get the minimum value of f(x,y) = f(5,5) = 25, i.e, taking the lowest nos. in the domain.
Sometimes you do post easy peasy cute questions!! :)
as we have product as a output and positive as well as negative values for x & y................ans must be negative.................and largest negative value we can make is........-50...............using big number with positive sing for one place and big number with negative sign for other.......!!!!!!!!!!!
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Since x and y have both positive and negative values in their domain, the function g(x,y) will be minimum when one of them (i.e. x or y) is minimum and the second one is maximum. Therefore, minimum of g(x,y) will be the product of -5 and 10, which is equal to -50.
Absolute value of -50 is 5 0
That's the answer!