मराठी

Prove that the following function do not have maxima or minima: g(x) = logx - Mathematics

Advertisements
Advertisements

प्रश्न

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

g(x) = logx

बेरीज
Advertisements

उत्तर

Given function g(x) = log x

∴ g'(x) = `1/x, x > 0`

`g (x) = 1/x ne 0` for all x ∈ (0, ∞)

⇒ x ∈ R, g'(x) is never equal to zero.

Hence there is no highest or lowest value of g.

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 6: Application of Derivatives - Exercise 6.5 [पृष्ठ २३२]

APPEARS IN

एनसीईआरटी Mathematics Part 1 and 2 [English] Class 12
पाठ 6 Application of Derivatives
Exercise 6.5 | Q 4.2 | पृष्ठ २३२

व्हिडिओ ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्‍न

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.


Find the maximum and minimum value, if any, of the following function given by f(x) = (2x − 1)2 + 3. 


Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2


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:

f(x) = x2


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)`


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

h(x) = x3 + x2 + x + 1


Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.


Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.


Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`


Find the maximum area of an isosceles triangle inscribed in the ellipse  `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.


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


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.`


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.


 Find the point on the straight line 2x+3y = 6,  which is closest to the origin. 


Divide the number 30 into two parts such that their product is maximum.


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.


Divide the number 20 into two parts such that their product is maximum


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 function y = 1 + sin x is maximum, when x = ______ 


The minimum value of the function f(x) = 13 - 14x + 9x2 is ______


The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.


Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.


The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.


The function `"f"("x") = "x" + 4/"x"` has ____________.


Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.


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


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


Let f: R → R be a function defined by f(x) = (x – 3)n1(x – 5)n2, n1, n2 ∈ N. Then, which of the following is NOT true?


The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.


If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.


Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.


A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x and y = 30 – 2x. The largest area of such a rectangle 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 ______.


A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.


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:

f(x) `= x sqrt(1 - x), 0 < x < 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.

The shortest distance between the line y - x = 1and the curve x = y2 is


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×