Advertisements
Advertisements
प्रश्न
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
विकल्प
1
0
–1
2
Advertisements
उत्तर
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is –1.
Explanation:
f(x) = `(x^2 - 1)/(x^2 + 1) = (x^2 + 1 - 2)/(x^2 + 1) = 1 - 2/(x^2 + 1)`
Therefore, f(x) < 1 ∀x and ≥ –1 ......`(∵ 2/(x^2 + 1) ≤ 2)`
Therefore, –1 ≤ f(x) < 1
Hence, f(x) has minimum value −1 and also there is no maximum value.
Alter: We have
f'(x) = `((x^2 + 1)2x - (x^2 - 1)2x)/(x^2 + 1)^2 = (4x)/(x^2 + 1)^2`
f'(x) = 0
⇒ x = 0
Now, f"(x) = `((x^2 + 1)^2 4 - 4x.2(x^2 + 1)2x)/(x^2 + 1)^4`
= `((x^2 + 1)4 - 16x(x))/(x^2 + 1)^3`
= `(-12x^2 + 4)/(x^2 + 1)^3`
Therefore, f′′(0) > 0 and there is only one critical point that has minima. Hence, f(x) has the least value at x = 0
fmin = f(0) = –1/1 = –1
संबंधित प्रश्न
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
Find the maximum and minimum value, if any, of the following function given by g(x) = x3 + 1.
Find the maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 1.
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 3|
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 the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) =x^3, x in [-2,2]`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
Find two numbers whose sum is 24 and whose product is as large as possible.
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
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 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box
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?
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 : y = 5x3 + 2x2 – 3x.
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
Find the maximum and minimum of the following functions : f(x) = x log x
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
Show that among rectangles of given area, the square has least perimeter.
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
The function f(x) = x log x is minimum at x = ______.
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
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.
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.
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) 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. |
- If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
- Find the critical point of the function.
- 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.
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 p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
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 ______.
Read the following passage:
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
|
Based on the above information, answer the following questions:
- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
OR
(b) For maximum volume, h > r. State true or false and justify. (2)
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
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"`
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
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`
The shortest distance between the line y - x = 1and the curve x = y2 is
