Advertisements
Advertisements
प्रश्न
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
Advertisements
उत्तर
f(x) = `x^2 + 16/x`
∴ f'(x) = `2x - 16/x^2`
and f"(x) = `2 + 32/x^3`
Consider, f'(x) = 0
∴ `2x - 16/x^2` = 0
∴ 2x = `16/x^2`
∴ x3 = 8
∴ x = 2
The maximum value is 2.
f(x) = `x^2 + 16/x`
For x = 2
f''(2) = `2 + 32/2^3`
= `2 + 32/8`
= 2 + 4
= 6 > 0
∴ f(x) attains minimum value at x = 2
∴ Minimum value = f(2) = `(2)^2 + 16/2 = 4 + 8` = 12
∴ The function f(x) has minimum value 12 at x = 2.
संबंधित प्रश्न
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.
Find the maximum and minimum value, if any, of the following function given by g(x) = x3 + 1.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
`f(x) = xsqrt(1-x), x > 0`
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
Find the maximum and minimum values of x + sin 2x on [0, 2π].
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 altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3.`
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
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 \[\pi : (\pi + 2)\].
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.
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?
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 : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
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.
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.
Determine the maximum and minimum value of the following function.
f(x) = x log x
If f(x) = x.log.x then its maximum value is ______.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
If y = x3 + x2 + x + 1, then y ____________.
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle 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 ______.
A straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.
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)
What is an absolute maximum?
