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
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
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 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`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
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.`
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 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box
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.
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?
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)`.
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
Determine the maximum and minimum value of the following function.
f(x) = x log x
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
If f(x) = x.log.x then its maximum value is ______.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
The function `"f"("x") = "x" + 4/"x"` has ____________.
Read the following passage and answer the questions given below.
|
In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of `x^2/a^2 + y^2/b^2` = 1. |
- If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
- Find the critical point of the function.
- Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
OR
Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
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 ______.
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 ______.
A straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
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.
