Advertisements
Advertisements
प्रश्न
If x + y = 3 show that the maximum value of x2y is 4.
Advertisements
उत्तर
x + y = 3
∴ y = 3 – x
Let T = x2y = x2(3 – x) = 3x2 – x3
Differentiating w.r.t. x, we get
`"dT"/("d"x) = 6"x" - 3"x"^2` ....(i)
Again, differentiating w.r.t. x, we get
`("d"^2"T")/("d"x^2) = 6 - 6"x"` ...(ii)
Consider, `"dT"/("d"x) = 0`
∴ 6x – 3x2 = 0
∴ x = 2
For x = 2,
`(("d"^2"T")/"dx"^2)_(x = 2)` = 6 – 6(2)
= 6 – 12
= – 6 < 0
Thus, T, i.e., x2y is maximum at x = 2
For x = 2, y = 3 – x = 3 – 2 = 1
∴ Maximum value of T = x2y = (2)2(1) = 4
संबंधित प्रश्न
Find the maximum and minimum value, if any, of the following function given by h(x) = x + 1, x ∈ (−1, 1)
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
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]
What is the maximum value of the function sin x + cos x?
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
Find the maximum and minimum of the following functions : f(x) = `logx/x`
Divide the number 30 into two parts such that their product is maximum.
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.
Solve the following : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.
Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
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?
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`
The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
The maximum value of `(1/x)^x` is ______.
Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The function `"f"("x") = "x" + 4/"x"` has ____________.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
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 ______.
The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, is ______.
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
Complete the following activity to divide 84 into two parts such that the product of one part and square of the other is maximum.
Solution: Let one part be x. Then the other part is 84 - x
Letf (x) = x2 (84 - x) = 84x2 - x3
∴ f'(x) = `square`
and f''(x) = `square`
For extreme values, f'(x) = 0
∴ x = `square "or" square`
f(x) attains maximum at x = `square`
Hence, the two parts of 84 are 56 and 28.
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local minimum?
