मराठी

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

Advertisements
Advertisements

प्रश्न

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

बेरीज
Advertisements

उत्तर

We have, h(x) = x  + 1, 1, -1 < x < 1.

-1 < x < 1

= -1 + 1 < x + 1 < 1 + 1

= 0 < x + 1 < 2

Here, range of f = (0, 2)

∴ f has neither a maximum nor a minimum value.

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 2.5 | पृष्ठ २३२

व्हिडिओ ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्‍न

If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.


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 g(x) = x3 + 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].


At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?


It is given that at x = 1, the function x4− 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.


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


Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.


Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .


Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x


Find the maximum and minimum of the following functions : f(x) = x log x


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


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 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.


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?


Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x


A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?


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


The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.


Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.


The maximum value of `(1/x)^x` is ______.


Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/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.


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


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


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


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


The function `f(x) = x^3 - 6x^2 + 9x + 25` has


Divide 20 into two ports, so that their product is maximum.


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 ______.


The minimum value of 2sinx + 2cosx 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.


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"`


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.


Which statement gives the meaning of a local minimum at \[c\]?


What is an absolute minimum?


For a continuous function on a closed interval \[[a,b]\], which values must be compared to find absolute maxima and minima?


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×