Advertisements
Advertisements
рдкреНрд░рд╢реНрди
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Advertisements
рдЙрддреНрддрд░
f'(x) = k(cos x - sin x) ...(given)
`f(x)=intf'(x)dx`
`=intk(cosx-sinx)dx`
`=kint(cosx-sinx)dx`
f(x) = k (sinx + cosx) + c ...(i)
f'(0) = 3 ...(given)
k(cos0 - sin0) = 3
k(1) = 3
k = 3 ...(ii)
also, `f(pi/2)=15`
`k[sin(pi/2)+cos(pi/2)]+c=15`
3(1 + 0) + c = 15
c = 12
Putting (ii) and (iii) in (i), we get
f(x) = 3(sinx + cosx) + 12
APPEARS IN
рд╕рдВрдмрдВрдзрд┐рдд рдкреНрд░рд╢реНтАНрди
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
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
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 absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
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?
Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
- local maxima
- local minima
- point of inflexion
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
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 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 wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
The total cost of producing x units is тВ╣ (x2 + 60x + 50) and the price is тВ╣ (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.
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 z = ax + by; a, b > 0 subject to x ≤ 2, y ≤ 2, x + y ≥ 3, x ≥ 0, y ≥ 0 has minimum value at (2, 1) only, then ______.
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
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`
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
The maximum value of the function f(x) = `logx/x` is ______.
A function f(x) is maximum at x = a when f'(a) > 0.
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.
The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° 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.
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.
