Advertisements
Advertisements
Question
What is the maximum value of the function sin x + cos x?
Advertisements
Solution
Let f(x) = sin x + cos x, be the interval [0, 2π]
f'(x) = cos x - sin x
For the highest and lowest values,
f'(x) = 0 ⇒ cos x - sin x = 0 ⇒ tan x = 1
`therefore x = pi/4 , (5 pi)/4`
Putting the values of x in f(x) = sin x + cos x, respectively,
At x = 0, f(0) = sin 0 + cos 0 = 1
x `= 2 pi "at," f(2 pi) = sin 2 pi + cos 2 pi = 1`
x`= pi /4 "at," f(pi/4) = sin pi/4 + cos pi/4`
`= 1/sqrt2 + 1/sqrt2 = 2/sqrt2`
`= sqrt2`
x `= (5pi)/4 "at," f((5pi)/4) `
`= sin (5 pi)/4 + cos (5 pi)/4`
`= - 1/sqrt2 + 1/sqrt2`
`= - 2/sqrt2`
`= -sqrt2`
Hence, the highest value is at x = `pi/4` = `sqrt2`.
APPEARS IN
RELATED QUESTIONS
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
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) = (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) = x/2 + 2/x, x > 0`
Prove that the following function do not have maxima or minima:
f(x) = ex
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) =x^3, x in [-2,2]`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
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.
Find two numbers whose sum is 24 and whose product is as large as possible.
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
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
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
Find the maximum and minimum of the following functions : f(x) = x log x
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Divide the number 20 into two parts such that their product 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.
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
The maximum value of `(1/x)^x` is ______.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
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.
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.
The minimum value of 2sinx + 2cosx is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
Read the following passage:
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
|
Based on the above information, answer the following questions:
- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
OR
(b) For maximum volume, h > r. State true or false and justify. (2)
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.
Check whether the function f : R `rightarrow` R defined by f(x) = x3 + x, has any critical point/s or not ? If yes, then find the point/s.
Sumit has bought a closed cylindrical dustbin. The radius of the dustbin is ‘r' cm and height is 'h’ cm. It has a volume of 20π cm3.
- Express ‘h’ in terms of ‘r’, using the given volume.
- Prove that the total surface area of the dustbin is `2πr^2 + (40π)/r`
- Sumit wants to paint the dustbin. The cost of painting the base and top of the dustbin is ₹ 2 per cm2 and the cost of painting the curved side is ₹ 25 per cm2. Find the total cost in terms of ‘r’, for painting the outer surface of the dustbin including the base and top.
- Calculate the minimum cost for painting the dustbin.
