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
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
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
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 x + y = 60 and xy3 is maximum.
Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
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
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
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.
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.
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
If f(x) = x.log.x then its maximum value is ______.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
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 ______.
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
If x is real, the minimum value of x2 – 8x + 17 is ______.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
The maximum value of `(1/x)^x` is ______.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
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 combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
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 maximum value of the function f(x) = `logx/x` is ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
The minimum value of 2sinx + 2cosx 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 ______.
The rectangle has area of 50 cm2. Complete the following activity to find its dimensions for least perimeter.
Solution: Let x cm and y cm be the length and breadth of a rectangle.
Then its area is xy = 50
∴ `y =50/x`
Perimeter of rectangle `=2(x+y)=2(x+50/x)`
Let f(x) `=2(x+50/x)`
Then f'(x) = `square` and f''(x) = `square`
Now,f'(x) = 0, if x = `square`
But x is not negative.
∴ `x = root(5)(2) "and" f^('')(root(5)(2))=square>0`
∴ by the second derivative test f is minimum at x = `root(5)(2)`
When x = `root(5)(2),y=50/root(5)(2)=root(5)(2)`
∴ `x=root(5)(2) "cm" , y = root(5)(2) "cm"`
Hence, rectangle is a square of side `root(5)(2) "cm"`
If x + y = 8, then the maximum value of x2y is ______.
Divide the number 100 into two parts so that the sum of their squares is minimum.
Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.
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.
