Advertisements
Advertisements
प्रश्न
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
उत्तर
\[\text { Let the two positive numbers be x and y}. \text{ Then, }\]
\[x + y = 15 ........ \left( 1 \right)\]
\[\text{Now}, \]
\[z = x^2 + y^2 \]
\[ \Rightarrow z = x^2 + \left( 15 - x \right)^2 ..........\left[ \text { From eq } . \left( 1 \right) \right]\]
\[ \Rightarrow z = x^2 + x^2 + 225 - 30x\]
\[ \Rightarrow z = 2 x^2 + 225 - 30x\]
\[ \Rightarrow \frac{dz}{dx} = 4x - 30\]
\[\text { For maximum or minimum values of z, we must have }\]
\[\frac{dz}{dx} = 0\]
\[ \Rightarrow 4x - 30 = 0\]
\[ \Rightarrow x = \frac{15}{2}\]
\[\frac{d^2 z}{d x^2} = 4 > 0\]
\[\text { Substituting x } = \frac{15}{2} \text{ in }\left( 1 \right), \text { we get } \]
\[y = \frac{15}{2}\]
\[\text { Thus, z is minimum when x = y } = \frac{15}{2} .\]
APPEARS IN
संबंधित प्रश्न
f(x) = - (x-1)2+2 on R ?
f(x) = | sin 4x+3 | on R ?
f(x) = x3 \[-\] 1 on R .
`f(x) = 2/x - 2/x^2, x>0`
`f(x) = x/2+2/x, x>0 `.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
How should we choose two numbers, each greater than or equal to `-2, `whose sum______________ so that the sum of the first and the cube of the second is minimum?
Given the sum of the perimeters of a square and a circle, show that the sum of there areas is least when one side of the square is equal to diameter of the circle.
A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimension of the rectangle so that its area is maximum. Find also the area ?
Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
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).
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.
The space s described in time t by a particle moving in a straight line is given by S = \[t5 - 40 t^3 + 30 t^2 + 80t - 250 .\] Find the minimum value of acceleration.
Write necessary condition for a point x = c to be an extreme point of the function f(x).
Write sufficient conditions for a point x = c to be a point of local maximum.
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
The number which exceeds its square by the greatest possible quantity is _________________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
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.
