Advertisements
Advertisements
प्रश्न
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
Advertisements
उत्तर
Let the length and breadth of a rectangle be l and b respectively.
∴ Perimeter of rectangle = 2(l + b) = 108 m
∴ l + b = 54
∴ b = 54 – l ...(i)
Area of rectangle = l × b
= l(54 – l) ...[From (i)]
Let f(l) = l(54 – l)
= 54l – l2
∴ f'(l) = 54 – 2l
∴ f"(l) = – 2
Consider, f'(l) = 0
∴ 54 – 2l = 0
∴ 54 = 2l
∴ l = 27
For l = 27,
f"(27) = – 2 < 0
∴ f(l) i.e., area is maximum at l = 27
and b = 54 – 27 ...[From (i)]
= 27
∴ The dimensions of rectangle are 27 m × 27 m.
संबंधित प्रश्न
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = sinx − cos x, 0 < x < 2π
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x3 − 6x2 + 9x + 15
Prove that the following function do not have maxima or minima:
f(x) = ex
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
Show that among rectangles of given area, the square has least perimeter.
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Solve the following : A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
If x + y = 3 show that the maximum value of x2y is 4.
The function f(x) = x log x is minimum at x = ______.
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
The maximum value of sin x . cos x is ______.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.
The minimum value of 2sinx + 2cosx is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) `= x sqrt(1 - x), 0 < x < 1`
