Advertisements
Advertisements
Question
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
Solution
\[\text { Let the two positive numbers be x and y}. \text{ Then, }\]
\[x + y = 15 ........ \left( 1 \right)\]
\[\text{Now}, \]
\[z = x^2 + y^2 \]
\[ \Rightarrow z = x^2 + \left( 15 - x \right)^2 ..........\left[ \text { From eq } . \left( 1 \right) \right]\]
\[ \Rightarrow z = x^2 + x^2 + 225 - 30x\]
\[ \Rightarrow z = 2 x^2 + 225 - 30x\]
\[ \Rightarrow \frac{dz}{dx} = 4x - 30\]
\[\text { For maximum or minimum values of z, we must have }\]
\[\frac{dz}{dx} = 0\]
\[ \Rightarrow 4x - 30 = 0\]
\[ \Rightarrow x = \frac{15}{2}\]
\[\frac{d^2 z}{d x^2} = 4 > 0\]
\[\text { Substituting x } = \frac{15}{2} \text{ in }\left( 1 \right), \text { we get } \]
\[y = \frac{15}{2}\]
\[\text { Thus, z is minimum when x = y } = \frac{15}{2} .\]
APPEARS IN
RELATED QUESTIONS
f(x) = - (x-1)2+2 on R ?
f(x)=sin 2x+5 on R .
f(x) = | sin 4x+3 | on R ?
f(x)=2x3 +5 on R .
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = (x \[-\] 5)4.
f(x) = (x \[-\] 1) (x+2)2.
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) = x4 \[-\] 62x2 + 120x + 9.
f(x) = xex.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
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?
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?
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.
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
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 coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
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.
The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.
The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere is ______________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
Which of the following graph represents the extreme value:-
