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
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of 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 function given by f(x) = |x + 2| − 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:
g(x) = x3 − 3x
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 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) = x3 − 6x2 + 9x + 15
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
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.
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
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.`
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
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.
The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?
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 ______.
Determine the maximum and minimum value of the following function.
f(x) = x log x
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
Divide the number 20 into two parts such that their product is maximum.
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
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?
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/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.
The maximum value of sin x . cos x is ______.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)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.
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
The function `"f"("x") = "x" + 4/"x"` has ____________.
A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then `(4/π + 1)`k is equal to ______.
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 the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
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 ______.
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
