Advertisements
Advertisements
Question
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Advertisements
Solution
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
RELATED QUESTIONS
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 profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
A square piece of tin of side 18 cm is to made into a box without a top by cutting a square from each corner and folding up the flaps to form the box. 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 altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3.`
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
Show that among rectangles of given area, the square has least perimeter.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
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
If f(x) = x.log.x then its maximum value is ______.
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, 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 ______.
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
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.
If x is real, the minimum value of x2 – 8x + 17 is ______.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The maximum value of `(1/x)^x` is ______.
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
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 ______.
The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
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 ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
The minimum value of 2sinx + 2cosx is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, is ______.
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
