Advertisements
Advertisements
Question
The maximum value of the function f(x) = `logx/x` is ______.
Options
e
`1/e`
e2
`1/e^2`
Advertisements
Solution
The maximum value of the function f(x) = `logx/x` is `bb(1/e)`.
APPEARS IN
RELATED QUESTIONS
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
g(x) = x3 − 3x
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
`g(x) = 1/(x^2 + 2)`
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
Find two numbers whose sum is 24 and whose product is as large as possible.
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
Find the maximum and minimum of the following functions : f(x) = `logx/x`
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.
The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
Show that among rectangles of given area, the square has least perimeter.
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.
Solve the following:
A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
If x + y = 3 show that the maximum value of x2y is 4.
Divide the number 20 into two parts such that their product is maximum
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
A metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.
Solution: Let the dimensions of the rectangle be x cm and y cm.
∴ 2x + 2y = 36
Let f(x) be the area of rectangle in terms of x, then
f(x) = `square`
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme value, f'(x) = 0, we get
x = `square`
∴ f''`(square)` = – 2 < 0
∴ Area is maximum when x = `square`, y = `square`
∴ Dimensions of rectangle are `square`
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
If y `= "ax - b"/(("x" - 1)("x" - 4))` has a turning point P(2, -1), then find the value of a and b respectively.
If y = x3 + x2 + x + 1, then y ____________.
A function f(x) is maximum at x = a when f'(a) > 0.
The sum of all the local minimum values of the twice differentiable function f : R `rightarrow` R defined by
f(x) = `x^3 - 3x^2 - (3f^('')(2))/2 x + f^('')(1)`
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, is ______.
Complete the following activity to divide 84 into two parts such that the product of one part and square of the other is maximum.
Solution: Let one part be x. Then the other part is 84 - x
Letf (x) = x2 (84 - x) = 84x2 - x3
∴ f'(x) = `square`
and f''(x) = `square`
For extreme values, f'(x) = 0
∴ x = `square "or" square`
f(x) attains maximum at x = `square`
Hence, the two parts of 84 are 56 and 28.
20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are
The shortest distance between the line y - x = 1and the curve x = y2 is
A maximum or minimum value of \[f\] is called a(n) _____.
When is a point in the domain of a function called a critical point?
Which statement gives the meaning of a local maximum at \[c\]?
For a continuous function on a closed interval \[[a,b]\], which values must be compared to find absolute maxima and minima?
Why must the left endpoint \[a\] and right endpoint \[b\] be checked for absolute extrema on \[[a,b]\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], which is the correct factorization of \[f'(x)\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=1\]?
