Advertisements
Advertisements
प्रश्न
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
Advertisements
उत्तर १
y = 5x3 + 2x2 – 3x
∴ `dy/dx = d/dx(5x^3 + 2x^2 - 3x)`
= 5 × 3x2 + 2 × 2x – 3 × 1
= 15x2 + 4x – 3
and `(d^2y)/(dx^2) = d/dx(15x^2 + 4x - 3)`
= 15 × 2x + 4 x 1 – 0
= 30x + 4
`dy/dx` = 0 gives 15x2 + 4x – 3 = 0
∴ 15x2 + 9x – 5x – 3 = 0
∴ 3x(5x + 3) – 1(5x + 3) = 0
∴ (5x + 3)(3x – 1) = 0
∴ x = `-(3)/(5) or x = (1)/(3)`
∴ the roots of `dy/dx = 0` are `x_1 = -(3)/(5) and x_2 = (1)/(3)`.
Second Derivative Test:
(a) `((d^2y)/dx^2)_("at" x = - 3/5) = 30(-3/5) + 4` = – 14 < 0
∴ by the second derivative test, y is maximum at x = `-(3)/(5)` and maximum value of y at x = `-(3)/(5)`
= `5(-3/5)^3 + 2(-3/5)^2 - 3(-3/5)`
= `(-27)/(25) + (18)/(25) + (9)/(5)`
= `(36)/(25)`
(b) `((d^2y)/dx^2)_("at" x = 1/3) = 30(1/3) + 4` = 14 > 0
∴ by the second derivative test, y is minimum at x = `(1)/(3)` and minimum value of y at x = `(1)/(3)`
= `5(1/3)^3 + 2(1/3)^2 - 3(1/3)`
= `(5)/(27) + (2)/(9) - 1`
= `-(16)/(27)`
Hence, the function has maximum value `(36)/(25)` at x = `-(3)/(5)` and minimum value `-(16)/(27) "at" x = (1)/(3)`.
उत्तर २
y = 5x3 + 2x2 – 3x
∴ `dy/dx = d/dx(5x^3 + 2x^2 - 3x)`
= 5 × 3x2 + 2 × 2x – 3 × 1
= 15x2 + 4x – 3
`dy/dx` = 0 gives 15x2 + 4x – 3 = 0
∴ 15x2 + 9x – 5x – 3 = 0
∴ 3x(5x + 3) – 1(5x + 3) = 0
∴ (5x + 3)(3x – 1) = 0
∴ x = `-(3)/(5) or x = (1)/(3)`
∴ the roots of `dy/dx = 0` are `x_1 = -(3)/(5) and x_2 = (1)/(3)`.
First Derivative Test:
(a) `dy/dx` = 15x2 + 4x – 3 = (5x + 3)(3x – 1)
Consider x = `-(3)/(5)`
Let h be a small positive number. Then
`(dy/dx)_("at" x = -3/5 - h) = [5(-3/5 - h) + 3][3 (-3/5 - h) - 1]`
= `( - 3 - 5h + 3)(-9/5 - 3h - 1)`
= `-5h (-14/5 - 3h)`
= `5h (14/5 + 3h) > 0`
and `(dy/dx)_("at" x = -3/5 + h) = [5(-3/5 + h) + 3][3 (-3/5 + h) - 1]`
= `(- 3 + 5h + 3)(-9/5 + 3h - 1)`
= `5h(3h - 14/5) < 0`,
as h is small positive number.
∴ by the first derivative test, y is maximum at x = `-(3)/(5)` and maximum value of y at x = `-(3)/(5)`
= `5(-3/5)^3 + 2(-3/5)^2 - 3(-3/5)`
= `-(27)/(25) + (18)/(25) + (9)/(5) = (36)/(25)`
(b) `dy/dx` = 15x2 + 4x – 3 = (5x + 3)(3x – 1)
Consider x = `(1)/(3)`
Let h be a small positive number. Then
`(dy/dx)_("at" x =1/3 - h) = [5(1/3 - h) + 3][3 (1/3 - h) - 1]`
= `(5/3 - 5h + 3)(1 - 3h - 1)`
= `(14/5 - 5h)(-3h)` < 0, as h is small positive number
and `(dy/dx)_("at" x = 1/3 + h) = [5(1/3 + h) + 3][3 (1/3 + h) - 1]`
= `(5/3 + 5h + 3)(1 + 3h - 1)`
= `(14/3 + 5h)(3h) > 0`
∴ by the first derivative test, y is minimum at x = `(1)/(3)` and minimum value of y at x = `(1)/(3)`
= `5(1/3)^3 + 2(1/3)^2 - 3(1/3)`
= `(5)/(27) + (2)/(9) - 1`
= `(-16)/(27)`
Hence, the function has maximum value `(36)/(25) "at" x = -(3)/(5)` and minimum value `-(16)/(27) "at" x = (1)/(3)`.
APPEARS IN
संबंधित प्रश्न
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere
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 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:
f(x) = x2
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:
`g(x) = 1/(x^2 + 2)`
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
What is the maximum value of the function sin x + cos x?
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 the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
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 surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
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) = x3 – 9x2 + 24x
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.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.
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 log x
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 R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
The function y = 1 + sin x is maximum, when x = ______
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
If x is real, the minimum value of x2 – 8x + 17 is ______.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
Range of projectile will be maximum when angle of projectile is
Divide 20 into two ports, so that their product is maximum.
Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P 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 ______.
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` 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 ______.
A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x and y = 30 – 2x. The largest area of such a rectangle is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
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.
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.
Divide the number 100 into two parts so that the sum of their squares is minimum.
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
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.
20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are
