Advertisements
Advertisements
Question
The maximum value of sin x . cos x is ______.
Options
`1/4`
`1/2`
`sqrt(2)`
`2sqrt(2)`
Solution
The maximum value of sin x . cos x is `1/2`.
Explanation:
We have f(x) = sin x cos x
⇒ f(x) = `1/2 * 2 sin x cos x`
= `1/2 sin 2x`
f'(x) = `1/2 * 2 cos 2x`
⇒ f'(x) = cos 2x
Now for local maxima and local minima f'(x) = 0
∴ cos 2x = 0
2x = `("n" + 1) pi/2`, n ∈ I
⇒ x = `(2"n" + 1) pi/4`
∴ x = `pi/4, (3pi)/4` .....
f"(x) = – 2 sin 2x
`"f''"(x)_(x = pi/4)` = `-2 sin 2 * pi/4`
= `- 2 sin pi/2`
= – 2 < 0 maxima
`"f''"(x)_(x = (3pi)/4) = - 2 sin 2 * (3pi)/4`
= `-2 sin (3pi)/4`
= 2 > 0 minima
So f(x) is maximum at x = `pi/4`
∴ Maximum value of f(x) = `sin pi/4 * cos pi/4`
= `1/sqrt(2) * 1/sqrt(2)`
= `1/sqrt(2)`.
APPEARS IN
RELATED QUESTIONS
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:
`h(x) = sinx + cosx, 0 < x < pi/2`
Prove that the following function do not have maxima or minima:
g(x) = logx
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
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 ______.
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
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
Find the maximum and minimum of the following functions : f(x) = `logx/x`
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.
Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.
Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
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?
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?
Divide the number 20 into two parts such that their product 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 ______
The two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.
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?
The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
The maximum value of `(1/x)^x` is ______.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
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 ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
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 ______.
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 maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
If Mr. Rane order x chairs at the price p = (2x2 - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?
Solution: Let Mr. Rane order x chairs.
Then the total price of x chairs = p·x = (2x2 - 12x- 192)x
= 2x3 - 12x2 - 192x
Let f(x) = 2x3 - 12x2 - 192x
∴ f'(x) = `square` and f''(x) = `square`
f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0
∴ f is minimum when x = 8
Hence, Mr. Rane should order 8 chairs for minimum cost of deal.
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.
