Advertisements
Advertisements
प्रश्न
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
विकल्प
1
2
\[2\frac{1}{2}\]
\[3\frac{1}{3}\]
Advertisements
उत्तर
2
\[\text { Given }: xy = 1\]
\[ \Rightarrow y = \frac{1}{x}\]
\[f\left( x \right) = x + \frac{1}{x}\]
\[ \Rightarrow f'\left( x \right) = 1 - \frac{1}{x^2}\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 1 - \frac{1}{x^2} = 0\]
\[ \Rightarrow x^2 - 1 = 0\]
\[ \Rightarrow x^2 = 1\]
\[ \Rightarrow x = \pm 1\]
\[ \Rightarrow x = 1 ..............\left( \text { Given }: x > 1 \right)\]
`rArry=1`
\[\text { Now,} \]
\[f''\left( x \right) = \frac{2}{x^3}\]
\[ \Rightarrow f''\left( 1 \right) = 2 > 0\]
\[\text { So, x = 1 is a local minima } . \]
\[ \therefore \text { Minimum value of } f\left( x \right) = f\left( 1 \right) = 1 + 1 = 2\]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = (x \[-\] 5)4.
f(x) = sin 2x, 0 < x < \[\pi\] .
f(x) = cos x, 0 < x < \[\pi\] .
f(x) = x4 \[-\] 62x2 + 120x + 9.
f(x) = xex.
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
`f(x)=xsqrt(1-x), x<=1` .
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
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 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?
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
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 \[\theta\] = \[\frac{\pi}{6}\] .
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 maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
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 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 ?
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?
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.
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)`.
The maximum value of x1/x, x > 0 is __________ .
For the function f(x) = \[x + \frac{1}{x}\]
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
The point on the curve y2 = 4x which is nearest to, the point (2,1) is _______________ .
If x+y=8, then the maximum value of xy is ____________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
