मराठी

Find the Local Maxima and Local Minima, If Any, of the Following Functions. Find Also the Local Maximum and the Local Minimum Values, as the Case May Be: `F(X) = Xsqrt(1-x), X > 0` - Mathematics

Advertisements
Advertisements

प्रश्न

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

`f(x) = xsqrt(1-x), x > 0`

Advertisements

उत्तर

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 3.8 | पृष्ठ २३२

व्हिडिओ ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्‍न

Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x). 


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


Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].


A square piece of tin of side 18 cm is to made into a box without a top  by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?


The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.


A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.

Show that the minimum length of the hypotenuse is `(a^(2/3) + b^(2/3))^(3/2).`


Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base. 


A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area. 


Find the maximum and minimum of the following functions : f(x) = `logx/x`


Divide the number 20 into two parts such that sum of their squares is minimum.


Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.


Show that among rectangles of given area, the square has least perimeter.


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


Determine the maximum and minimum value of the following function.

f(x) = `x^2 + 16/x`


A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum


A metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.

Solution: Let the dimensions of the rectangle be x cm and y cm.

∴ 2x + 2y = 36

Let f(x) be the area of rectangle in terms of x, then

f(x) = `square`

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme value, f'(x) = 0, we get

x = `square`

∴ f''`(square)` = – 2 < 0

∴ Area is maximum when x = `square`, y = `square`

∴ Dimensions of rectangle are `square`


If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.


If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.


Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`


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.


Find all the points of local maxima and local minima of the function f(x) = (x - 1)(x + 1)2


Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < 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.


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


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 ____________.


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


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


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 cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.


Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.


The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.


If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).


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.


If Mr. Rane order x chairs at the price p = (2x2 - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?

Solution: Let Mr. Rane order x chairs.

Then the total price of x chairs = p·x = (2x2 - 12x- 192)x

= 2x3 - 12x2 - 192x

Let f(x) = 2x3 - 12x2 - 192x

∴ f'(x) = `square` and f''(x) = `square`

f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0

∴ f is minimum when x = 8

Hence, Mr. Rane should order 8 chairs for minimum cost of deal.


Find the maximum and the minimum values of the function f(x) = x2ex.


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.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×