मराठी
महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता १२ वी

The maximum value of the function f(x) = logxx is ______.

Advertisements
Advertisements

प्रश्न

The maximum value of the function f(x) = `logx/x` is ______.

पर्याय

  • e

  • `1/e`

  • e2

  • `1/e^2`

MCQ
रिकाम्या जागा भरा
Advertisements

उत्तर

The maximum value of the function f(x) = `logx/x` is `bb(1/e)`.

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
2021-2022 (March) Set 1

व्हिडिओ ट्यूटोरियलVIEW ALL [2]

संबंधित प्रश्‍न

If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).


Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10 


Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

`f(x) = xsqrt(1-x), x > 0`


Prove that the following function do not have maxima or minima:

f(x) = ex


Prove that the following function do not have maxima or minima:

g(x) = logx


Prove that the following function do not have maxima or minima:

h(x) = x3 + x2 + x + 1


Find the absolute maximum value and the absolute minimum value of the following function in the given interval:

f (x) = sin x + cos x , x ∈ [0, π]


Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].


Find the maximum and minimum values of x + sin 2x on [0, 2π].


A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening


A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.

Show that the minimum length of the hypotenuse is `(a^(2/3) + b^(2/3))^(3/2).`


Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.


Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.


 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. 


Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`


A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.


Solve the following:

A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.


Determine the maximum and minimum value of the following function.

f(x) = 2x3 – 21x2 + 36x – 20


Determine the maximum and minimum value of the following function.

f(x) = x log x


A metal wire of  36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.


If f(x) = x.log.x then its maximum value is ______.


By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima

Solution: f(x) = x3 – 9x2 + 24x

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme values, f'(x) = 0, we get

x = `square` or `square`

∴ f''`(square)` = – 6 < 0

∴ f(x) is maximum at x = 2.

∴ Maximum value = `square`

∴ f''`(square)` = 6 > 0

∴ f(x) is maximum at x = 4.

∴ Minimum value = `square`


The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.


The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.


Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______ 


If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.


The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______


A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.


The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `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 the sphere. Also find the minimum value of the sum of their volumes.


If x is real, the minimum value of x2 – 8x + 17 is ______.


The maximum value of sin x . cos x is ______.


Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.


The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:


Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.


If y = x3 + x2 + x + 1, then y ____________.


The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.


The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.


A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)


The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is


For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`


The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is


Read the following passage and answer the questions given below.

In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of `x^2/a^2 + y^2/b^2` = 1.

  1. If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
  2. Find the critical point of the function.
  3. Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
    OR
    Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.

If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.


The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.


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 ______.


Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).


Sumit has bought a closed cylindrical dustbin. The radius of the dustbin is ‘r' cm and height is 'h’ cm. It has a volume of 20π cm3.

  1. Express ‘h’ in terms of ‘r’, using the given volume.
  2. Prove that the total surface area of the dustbin is `2πr^2 + (40π)/r`
  3. Sumit wants to paint the dustbin. The cost of painting the base and top of the dustbin is ₹ 2 per cm2 and the cost of painting the curved side is ₹ 25 per cm2. Find the total cost in terms of ‘r’, for painting the outer surface of the dustbin including the base and top.
  4. Calculate the minimum cost for painting the dustbin.

Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×