Maxima/Minima

In NutsAndBolts factory, one machine produces only nuts at the rate of 100 nuts per minute and needs to be cleaned for 5 minutes after production of every 1000 nuts. Another machine produces only bolts at the rate of 75 bolts per minute and needs to cleaned for10 minutes after production of every 1500 bolts. If both the machines start production at the same time, what is the minimum duration required for producing 9000 pairs of nuts and bolts?

Each family in a locality has at most two adults, and no family has fewer than 3 children. Considering all the families together, there are more adults than boys, more boys than girls, and more girls than families, then the minimum possible number of families in the locality is

The maximum possible value of y = min (1/2 – 3x²/4, 5x²/4) for the range 0 < x < 1 is

Direction for questions 58 to 87: Answer the questions independently.

Largest value of min(2 + x², 6 – 3x), when x > 0, is

If a, b, c and d are four positive real numbers such that abcd = 1, what is the minimum value of (1 + a)(1 + b)(1 + c)(1 + d)?

Let x and y be two positive numbers such that x + y = 1. Then the minimum value ofis

Using only 2, 5, 10, 25, and 50 paisa coins, what will be the minimum number of coins required to pay exactly 78 paise, 69 paise and Rs. 1.01 to three different persons?

If three positive real numbers x, y and z satisfy y – x = z – y and x y z = 4, then what is the minimum possible value of y?

A square tin sheet of side 12 inches is converted into a box with open top in the following steps. The sheet is placed horizontally. Then, equal-sized squares, each of side x inches, are cut from the four corners of the sheet. Finally, the four resulting sides are bent vertically upwards in the shape of a box. If x is an integer, then what value of x maximizes the volume of the box?

An elevator has a weight limit of 630 kg. It is carrying a group of people of whom the heaviest weighs 57 kg and the lightest weighs 53 kg. What is the maximum possible number of people in the group?

If a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of (a – b)² + (a – c)² + (a – d)² is

Let f(x) = min{2x²,52−5x}, where x is any positive real number. Then the maximum possible value of f(x) is

Let f(x)=max{5x, 52-2x²}, where x is any positive real number. Then the minimum possible value of f(x) is

Among 100 students, x₁ have birthdays in January, x₂ have birthdays in February, and so on. If x₀ = max (x₁, x₂, …, x₁₂), then the smallest possible value of x0 is

For natural numbers x, y, and z, if xy + yz = 19 and yz + xz = 51, then the minimum possible value of xyz is

If p² + q² – 29 = 2pq – 20 = 52 – 2pq, then the difference between the maximum and minimum possible value of (p³ – q³) is