Advertisements
Advertisements
प्रश्न
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
उत्तर
\[\text { Let the two numbers be x and y. Then },\]
\[x + y = 15 ............. (1)\]
\[\text { Now,} \]
\[ z = x^2 y^3 \]
\[ \Rightarrow z = x^2 \left( 15 - x \right)^3 ............\left[ \text { From eq } . \left( 1 \right) \right]\]
\[ \Rightarrow \frac{dz}{dx} = 2x \left( 15 - x \right)^3 - 3 x^2 \left( 15 - x \right)^2 \]
\[\text { For maximum or minimum values of z, we must have }\]
\[\frac{dz}{dx} = 0\]
\[ \Rightarrow 2x \left( 15 - x \right)^3 - 3 x^2 \left( 15 - x \right)^2 = 0\]
\[ \Rightarrow 2x\left( 15 - x \right) = 3 x^2 \]
\[ \Rightarrow 30x - 2 x^2 = 3 x^2 \]
\[ \Rightarrow 30x = 5 x^2 \]
\[ \Rightarrow x = 6 \text { and }y = 9\]
\[\frac{d^2 z}{d x^2} = 2 \left( 15 - x \right)^3 - 6x \left( 15 - x \right)^2 - 6x \left( 15 - x \right)^2 + 6 x^2 \left( 15 - x \right)\]
\[\text { At x } = 6: \]
\[\frac{d^2 z}{d x^2} = 2 \left( 9 \right)^3 - 36 \left( 9 \right)^2 - 36 \left( 9 \right)^2 + 6\left( 36 \right)\left( 9 \right)\]
\[ \Rightarrow \frac{d^2 z}{d x^2} = - 2430 < 0\]
\[\text { Thus, z is maximum when x = 6 and y = 9 } . \]
\[\text { So, the required two parts into which 15 should be divided are 6 and 9 } .\]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f(x) = x3 (x \[-\] 1)2 .
f(x) = (x \[-\] 1) (x+2)2.
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
`f(x)=xsqrt(1-x), x<=1` .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
Find the maximum and minimum values of y = tan \[x - 2x\] .
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Find the absolute maximum and minimum values of a function f given by \[f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]\] ?
A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
Find the point at which M is maximum in a given case.
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the circle and the square is minimum?
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.
A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] Find the daily output to maximum the total profit.
Manufacturer can sell x items at a price of rupees \[\left( 5 - \frac{x}{100} \right)\] each. The cost price is Rs \[\left( \frac{x}{5} + 500 \right) .\] Find the number of items he should sell to earn maximum profit.
An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.
A particle is moving in a straight line such that its distance at any time t is given by S = \[\frac{t^4}{4} - 2 t^3 + 4 t^2 - 7 .\] Find when its velocity is maximum and acceleration minimum.
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the point where f(x) = x log, x attains minimum value.
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum?
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
