Advertisements
Advertisements
प्रश्न
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Advertisements
उत्तर
Given function, h(x) = x3 + x2 + x + 1
∴ h'(x) = 3x2 + 2x + 1 x ∈ R.
if, 3x2 + 2x + 1 = 0 then,
x `= (-2 pm sqrt(4 - 12))/6`
x `= (-2 pm sqrt(-8))/6`
`= (-1 pm sqrt (-2))/3` ...[which is non-real]
For x ∈ R, h'(x) ≠ 0
Hence there is no highest or lowest value of h.
APPEARS IN
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.
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 and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
Find the maximum and minimum value, if any, of the following function given by h(x) = x + 1, x ∈ (−1, 1)
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
Prove that the following function do not have maxima or minima:
f(x) = ex
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) =x^3, x in [-2,2]`
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
Find the maximum area of an isosceles triangle inscribed in the ellipse `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
Find the point on the straight line 2x+3y = 6, which is closest to the origin.
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
Find the maximum and minimum of the following functions : f(x) = x log x
The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?
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^2 + 16/x`
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 ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is 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`
The function y = 1 + sin x is maximum, when x = ______
The two parts of 120 for which the sum of double of first and square of second part is minimum, 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?
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
Range of projectile will be maximum when angle of projectile is
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, 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 ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` 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 ______.
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.
