minimum and maximum value?

Question

Let

a=3-5x
b=2x-4
c=1-4x

y=min(a,b,c) for different values of x.

The maximum value of y is ...

(1)5/6
(2)-7/3
(3)-17/6
(4)-7/6


I did not understand this question even.

see, they are saying y=min(a,b,c)
and again asking for max of y ?

its confusing...BTW, what does it mean by min(a,b,c) ? whats this function min()?

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hi, i understand the question now.

but calculation of maximum value of y is difficult.

is there any short cut method ? easy way to find it ?

Answer 1

For example, when x=1, a=-2, b=-2, and c=-3. The smallest value is -3, so y(1)=-3.

Again, if x=0, a=3, b=-4, and c=1, so y(0)=-4.

The problem is asking what the largest value for y will be as x changes. You might start by graphing the three lines represented by a,b, and c. Then, at each x value, the y value for this function will be the smallest of the three lines. Plot this out for every value of x and see where the largest y value occurs and what it is.

Answer 2

Let me add on to mathematician's explanation.

Generally, if f(x) and g(x) are two continuous functions, the function min(f(x),g(x)) and max(f(x),g(x)) are also continuous functions. Given a value of x in the domain of f(x) and g(x), min(f(x),g(x)) is the smaller of the two values.

Thus, in this case, min(3-5x,2x-4,1-4x) is a continuous function so on any _closed_ interval it will have a maxium and minimum value. However, on its entire domain it may or may not have an absolute maximum or an absolute minimum.

Let's see how it looks. After graphing each line, let's find the points of intersection.
Notice that y=2x-4 and y=1-4x intersect if x=5/6. Also, y=3-5x and y=1-4x intersect if x=2.

Thus, min(3-5x,2x-4,1-4x)=
2x-4 if x <=5/6
1-4x if 5/6 < x <=2
3-5x if x>2

Observe that the maximum of this function occurs if x=5/6.

If x=5/6, 2x-4=3-4x=-7/3.

On its entire domain, the function has a maximum value of -7/3. However, it does not have a minimum value on its entire domain.

Answer 3

For a given value of x, a,b and c will have certain values.

y 's value is the minimum among a,b,c

min() finds the minimum value among the given arguments.

Of all possible values of y, you now have to say, which will be the maximum value.