Advertisements
Advertisements
प्रश्न
Find the maximum and minimum value, if any, of the following function given by f(x) = (2x − 1)2 + 3.
Advertisements
उत्तर
We have, f(x) = (2x - 1)2 + 3 for all x ∈ R.
Since, (2x - 1)2 ≥ 0
= (2x - 1)2 + 3 ≥ 3
∴ Minimum f (x) = 3, which occurs when 2x - 1 = 0 i.e, when x = `1/2`
Value of f (x) has no maximum value, because f (x) → ∞ as |x| → ∞
APPEARS IN
संबंधित प्रश्न
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`. Also find maximum volume in terms of volume of the sphere
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
Find the maximum and minimum value, if any, of the following function given by g(x) = x3 + 1.
Find the maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 1.
What is the maximum value of the function sin x + cos x?
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
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.
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?
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.
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.
An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.
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 a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
If x + y = 3 show that the maximum value of x2y is 4.
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
The function y = 1 + sin x is maximum, when x = ______
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
An open box with square base is to be made of a given quantity of cardboard of area c2. Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units
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/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
The maximum value of the function f(x) = `logx/x` is ______.
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 ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° 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 two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
Find the maximum and the minimum values of the function f(x) = x2ex.
Divide the number 100 into two parts so that the sum of their squares is minimum.
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) `= x sqrt(1 - x), 0 < x < 1`
