Advertisements
Advertisements
Question
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
Advertisements
Solution
Given function h(x) = sin (2x) + 5
We know that -1 ≤ sin 2x ≤ 1
⇒ 4 ≤ 5 + sin 2x ≤ 6
Maximum value of sin 2x = 1
∴ h(x) = Maximum value of sin 2x + 5, 1 + 5 = 6
Minimum value of sin 2x = - 1
∴ h(x) = Minimum value of sin 2x + 5 = -1 + 5 = 4
APPEARS IN
RELATED QUESTIONS
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
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 maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
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 maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.
Find the maximum and minimum of the following functions : f(x) = x log x
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 of all rectangles inscribed in a given circle, the square has the maximum area.
Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
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.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
A metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.
Solution: Let the dimensions of the rectangle be x cm and y cm.
∴ 2x + 2y = 36
Let f(x) be the area of rectangle in terms of x, then
f(x) = `square`
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme value, f'(x) = 0, we get
x = `square`
∴ f''`(square)` = – 2 < 0
∴ Area is maximum when x = `square`, y = `square`
∴ Dimensions of rectangle are `square`
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
The two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.
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`
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 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.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.
The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.
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 ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
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.
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.
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
