Advertisements
Advertisements
प्रश्न
For the function f(x) = \[x + \frac{1}{x}\]
विकल्प
x = 1 is a point of maximum
x = \[-\] 1 is a point of minimum
maximum value > minimum value
maximum value < minimum value
उत्तर
\[\text { maximum value < minimum value}\]
\[\text { Given:} f\left( x \right) = x + \frac{1}{x}\]
\[ \Rightarrow f'\left( x \right) = 1 - \frac{1}{x^2}\]
\[\text { For a local maxima or a local minima, we must have} \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 1 - \frac{1}{x^2} = 0\]
\[ \Rightarrow x^2 - 1 = 0\]
\[ \Rightarrow x^2 = 1\]
\[ \Rightarrow x = \pm 1\]
\[\text { Now }, \]
\[f''\left( x \right) = \frac{2}{x^3}\]
\[ \Rightarrow f''\left( 1 \right) = \frac{2}{1} = 2 > 0\]
\[\text { So, x = 1 is a local minima.}\]
\[\text { Also }, \]
\[f''\left( - 1 \right) = - 2 < 0\]
\[\text {So, x = - 1 is a localmaxima }.\]
\[\text { The local minimum value is given by }\]
\[f\left( 1 \right) = 2\]
\[\text { The local maximum value is given by }\]
\[f\left( - 1 \right) = - 2\]
\[ \therefore \text { Maximum value < Minimum value }\]
APPEARS IN
संबंधित प्रश्न
f(x)=sin 2x+5 on R .
f(x) = (x \[-\] 5)4.
f(x) = x3 \[-\] 6x2 + 9x + 15 .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
f(x) = x3\[-\] 6x2 + 9x + 15
f(x) = xex.
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
Find the absolute maximum and minimum values of a function f given by \[f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]\] ?
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle 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 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 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.
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.
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 ?
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
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 straight line is drawn through a given point P(1,4). Determine the least value of the sum of the intercepts on the coordinate axes ?
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.
Write sufficient conditions for a point x = c to be a point of local maximum.
Write the point where f(x) = x log, x attains minimum value.
Write the minimum value of f(x) = xx .
Write the maximum value of f(x) = x1/x.
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value 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.
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
