Advertisements
Advertisements
प्रश्न
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Advertisements
उत्तर
Let f (x) = x + sin2x, 0 ≤ x ≤ 2π
⇒ f' (x) = 1 + 2cos 2x
⇒ For critical points, let f' (x) = 0
⇒ 1 + cos 2x = 0
⇒ `cos 2x = -1/2`
⇒ `cos 2x = -cos pi/3`
(If 0< x < 2π, then 0< 2x < 4π)
`⇒ cos 2x = cos (pi- pi/3), cos (pi + pi/2), cos (3pi - pi/3), cos (3pi + pi/3)`
⇒ `2x = (2pi)/3 , (4pi)/3, (8pi)/3, (10pi)/3`
⇒ `x = pi/3, (2pi)/3, (4pi)/3, (5pi)/3`
So, for finding maximum and minimum, we evaluate f (x) at `0, 2pi , pi/3, (2pi)/3, (4pi)/3, (5pi)/3`
Now f(0) = 0 + sin 0 =
f (2π) = 2π + sin 4π = 2π + 0 = 2π
`f (pi/3) = pi/3 + sin (2pi)/3 = pi/3 + sin (pi - pi/3)`
= `pi/3 + sin pi/3 = pi/3 + sqrt3/2`
`f ((2pi)/3) = (2pi)/3 + sin (4pi)/3 = (2pi)/3 + sin (pi + pi/3)`
= `(2pi)/3 -sin pi/3 = (2pi)/3 - sqrt3/2`
`f((4pi)/3) = (4pi)/3 + sin (8pi)/3 = (4pi)/3 + sin (2pi + (2pi)/3)`
= `(4pi)/3 + sin (2pi)/3 = (4pi)/3 + sqrt3/2`
and `f ((5pi)/3) = (5pi)/3 + sin (10pi)/3 = (5pi)/3 + sin (3pi + pi/3)`
= `(5pi)/3 -sin pi/3 = (5pi)/3 - sqrt3/2`
Thus, maximum value of f (x) = 2π at x = 2π and minimum value of f (x) = 0 at x = 0.
APPEARS IN
संबंधित प्रश्न
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
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) = sinx − cos x, 0 < x < 2π
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 maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
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.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
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) = 2x3 – 21x2 + 36x – 20
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Find the maximum and minimum of the following functions : f(x) = x log x
Divide the number 30 into two parts such that their product is maximum.
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.
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 : 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)`.
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The maximum value of `(1/x)^x` is ______.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
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 ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
The minimum value of 2sinx + 2cosx 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 ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other 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"`
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.
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`
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
What is an absolute minimum?
Let \[c\] be a critical point of a continuous function \[f\]. If \[f'(x)\] changes sign from positive to negative as \[x\] passes through \[c\], what is \[c\]?
