Advertisements
Advertisements
प्रश्न
Prove that the following function do not have maxima or minima:
g(x) = logx
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.
APPEARS IN
संबंधित प्रश्न
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
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 maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
Find the maximum and minimum value, if any, of the following function given by h(x) = x + 1, x ∈ (−1, 1)
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:
`h(x) = sinx + cosx, 0 < x < pi/2`
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
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
Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
- local maxima
- local minima
- point of inflexion
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Divide the number 30 into two parts such that their product is maximum.
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 : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.
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.
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.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
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 ______.
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
If y = x3 + x2 + x + 1, then y ____________.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
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)`
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then `(4/π + 1)`k is equal to ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.
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.
- Express ‘h’ in terms of ‘r’, using the given volume.
- Prove that the total surface area of the dustbin is `2πr^2 + (40π)/r`
- 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.
- Calculate the minimum cost for painting the dustbin.
The shortest distance between the line y - x = 1and the curve x = y2 is
For a function defined on an interval \[I\], which condition means that \[f\] has a maximum value at \[c\in I\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=0\]?
