Advertisements
Advertisements
प्रश्न
What is the maximum value of the function sin x + cos x?
उत्तर
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
संबंधित प्रश्न
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
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 function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x2
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:
h(x) = x3 + x2 + x + 1
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 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
Divide the number 30 into two parts such that their product is maximum.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
A metal wire of 36cm long is bent to form a rectangle. Find it's dimensions when it's area is maximum.
The function f(x) = x log x is minimum at x = ______.
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
The function y = 1 + sin x is maximum, when x = ______
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also find the maximum volume.
Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
The function `f(x) = x^3 - 6x^2 + 9x + 25` has
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
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.
The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.
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 ______.
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 ______.
If the point (1, 3) serves as the point of inflection of the curve y = ax3 + bx2 then the value of 'a ' and 'b' are ______.
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 minimum value of 2sinx + 2cosx 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 ______.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
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.
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.
