Advertisements
Advertisements
प्रश्न
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Advertisements
उत्तर
Let f(x) = sin 2x, interval [0, 2π]
f‘(x) = 2 cos 2x
यदि f'(x) = 0 ⇒ 2 cos 2x = 0
⇒ 2x `= pi/2, (3pi)/2, (5pi)/2, (7 pi)/2 => x = pi/4, (3pi)/4, (5pi)/4, (7 pi)/4`
Hence we find `x = pi/4, (3pi)/4, (5pi)/4, (7 pi)/4` and the value of f at the endpoints of the interval [0, 2 `pi`].
At x = 0, f (0) = sin 0 = 0
x `= 2 pi at, f(2 pi) = sin 2 xx 2 pi = sin 4 pi = 0`
x`= pi/4 at, f(pi/4) = sin 2 xxpi/4 = sin pi/2 = 1`
x `= (3pi)/4 at, f((3 pi)/4) = sin (3 pi)/2 = - 1`
x `= (5pi)/4 at, f((5pi)/4) = sin (5 pi)/2 = 1`
x `= (7pi)/4 at, f((7pi)/4) = sin (7 pi)/2 = -1`
Thus, the function f(x) attains maximum value 1 at `= pi/4` and x`= (5 pi)/4`.
APPEARS IN
संबंधित प्रश्न
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.
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:
f(x) = sinx − cos x, 0 < x < 2π
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) = sin x + cos x , x ∈ [0, π]
What is the maximum value of the function sin x + cos x?
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
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) = x3 – 9x2 + 24x
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Divide the number 30 into two parts such that their product is maximum.
A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?
Show that among rectangles of given area, the square has least perimeter.
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
Solve the following:
A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.
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.
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
The function `"f"("x") = "x" + 4/"x"` has ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
Range of projectile will be maximum when angle of projectile is
The maximum value of the function f(x) = `logx/x` is ______.
Divide 20 into two ports, so that their product is maximum.
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 ______.
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 minimum value of the function f(x) = xlogx is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, is ______.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
Divide the number 100 into two parts so that the sum of their squares is minimum.
