Advertisements
Advertisements
प्रश्न
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
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).
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) = −(x − 1)2 + 10
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 following function given by h(x) = sin(2x) + 5.
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)`
Prove that the following function do not have maxima or minima:
g(x) = logx
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
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.
Find two numbers whose sum is 24 and whose product is as large as possible.
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.
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.
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
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 cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.
Find the point on the straight line 2x+3y = 6, which is closest to the origin.
Divide the number 30 into two parts such that their product is maximum.
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.
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.
An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.
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.
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
Solve the following : A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
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 x + y = 3 show that the maximum value of x2y is 4.
Divide the number 20 into two parts such that their product is maximum
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area 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`
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 minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
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.
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.
If x is real, the minimum value of x2 – 8x + 17 is ______.
The maximum value of sin x . cos x is ______.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
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 maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
The maximum value of the function f(x) = `logx/x` is ______.
Let A = [aij] be a 3 × 3 matrix, where
aij = `{{:(1, "," if "i" = "j"),(-x, "," if |"i" - "j"| = 1),(2x + 1, "," "otherwise"):}`
Let a function f: R→R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to ______.
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 the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
If the point (1, 3) serves as the point of inflection of the curve y = ax3 + bx2 then the value of 'a ' and 'b' are ______.
Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b is ______.
The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.
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 ______.
The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.
The maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° is ______.
The minimum value of the function f(x) = xlogx is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
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.
Find the maximum and the minimum values of the function f(x) = x2ex.
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.
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
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`
