Advertisements
Advertisements
Question
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
Options
-2
0
3
none of these
Advertisements
Solution
none of these
\[\text { Given }: f\left( x \right) = x + \frac{1}{x}\]
\[ \Rightarrow f'\left( x \right) = 1 - \frac{1}{x^2}\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 1 - \frac{1}{x^2} = 0\]
\[ \Rightarrow x^2 - 1 = 0\]
\[ \Rightarrow x^2 = 1\]
\[ \Rightarrow x = \pm 1\]
\[ \Rightarrow x = 1 ................\left( \text { Given }: x>0 \right)\]
\[\text { Now,} \]
\[f''\left( x \right) = \frac{2}{x^3}\]
\[ \Rightarrow f''\left( 1 \right) = 2 > 0\]
\[\text { So, x = 1 is a local minima } .\]
APPEARS IN
RELATED QUESTIONS
f(x) = - (x-1)2+2 on R ?
f(x) = x3 \[-\] 3x.
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
Find the point of local maximum or local minimum, if any, of the following function, using the first derivative test. Also, find the local maximum or local minimum value, as the case may be:
f(x) = x3(2x \[-\] 1)3.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
Find the maximum and minimum values of y = tan \[x - 2x\] .
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
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 the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .
Find the absolute maximum and minimum values of a function f given by `f(x) = 12 x^(4/3) - 6 x^(1/3) , x in [ - 1, 1]` .
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
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?
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 \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
Find the point at which M is maximum in a given case.
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 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 the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
Determine the points on the curve x2 = 4y which are nearest 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 ?
A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?
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 necessary condition for a point x = c to be an extreme point of the function f(x).
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
Write the maximum value of f(x) = x1/x.
For the function f(x) = \[x + \frac{1}{x}\]
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum?
Which of the following graph represents the extreme value:-
