Advertisements
Advertisements
प्रश्न
The maximum value of sin x . cos x is ______.
विकल्प
`1/4`
`1/2`
`sqrt(2)`
`2sqrt(2)`
Advertisements
उत्तर
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
संबंधित प्रश्न
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 maximum and minimum value, if any, of the following function given by f(x) = (2x − 1)2 + 3.
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) = x3 − 6x2 + 9x + 15
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
What is the maximum value of the function sin x + cos x?
It is given that at x = 1, the function x4− 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
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?
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
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.
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
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?
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/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?
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
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.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
Let A = [aij] be a 3 × 3 matrix, where
aij = `{{:(1, "," if "i" = "j"),(-x, "," if |"i" - "j"| = 1),(2x + 1, "," "otherwise"):}`
Let a function f: R→R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to ______.
Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b is ______.
The sum of all the local minimum values of the twice differentiable function f : R `rightarrow` R defined by
f(x) = `x^3 - 3x^2 - (3f^('')(2))/2 x + f^('')(1)`
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
The minimum value of the function f(x) = xlogx is ______.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
