Advertisements
Advertisements
प्रश्न
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
Advertisements
उत्तर
\[\text { Suppose 64 is divided into two partsxand 64-x. Then, }\]
\[z = x^3 + \left( 64 - x \right)^3 \]
\[ \Rightarrow \frac{dz}{dx} = 3 x^2 + 3 \left( 64 - x \right)^2 \]
\[\text { For maximum or minimum values of z, we must have }\]
\[\frac{dz}{dx} = 0\]
\[ \Rightarrow 3 x^2 + 3 \left( 64 - x \right)^2 = 0\]
\[ \Rightarrow 3 x^2 = 3 \left( 64 - x \right)^2 \]
\[ \Rightarrow x^2 = x^2 + 4096 - 128x\]
\[ \Rightarrow x = \frac{4096}{128}\]
\[ \Rightarrow x = 32\]
\[\text { Now, }\]
\[\frac{d^2 z}{d x^2} = 6x + 6\left( 64 - x \right) \]
\[ \Rightarrow \frac{d^2 z}{d x^2} = 384 > 0\]
\[\text { Thus, z is minimum when 64 is divided into two equal parts, 32 and 32.}\]
APPEARS IN
संबंधित प्रश्न
f(x) = - (x-1)2+2 on R ?
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = x3 \[-\] 1 on R .
f(x) = x3 (x \[-\] 1)2 .
f(x) = sin 2x, 0 < x < \[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = (x - 1) (x + 2)2.
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
`f(x)=xsqrt(1-x), x<=1` .
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?
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 \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .
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?
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 tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?
A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 metres find the dimensions of the rectangle will produce the largest area of the window.
Prove that a conical tent of given capacity will require the least amount of canavas when the height is \[\sqrt{2}\] times the radius of the base.
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
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.
The total area of a page is 150 cm2. The combined width of the margin at the top and bottom is 3 cm and the side 2 cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?
Write sufficient conditions for a point x = c to be a point of local maximum.
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
If x+y=8, then the maximum value of xy is ____________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
The sum of the surface areas of a cuboid with sides x, 2x and \[\frac{x}{3}\] 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 sphere. Also find the minimum value of the sum of their volumes.
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
