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प्रश्न
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
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उत्तर
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of Local minima.
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संबंधित प्रश्न
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
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 points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
- local maxima
- local minima
- point of inflexion
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?
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
Divide the number 20 into two parts such that sum of their squares is minimum.
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Solve the following:
A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.
Determine the maximum and minimum value of the following function.
f(x) = x log 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?
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.
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
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 two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.
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.
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 all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
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 distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
Range of projectile will be maximum when angle of projectile 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 ______.
If the point (1, 3) serves as the point of inflection of the curve y = ax3 + bx2 then the value of 'a ' and 'b' are ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
The maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Complete the following activity to divide 84 into two parts such that the product of one part and square of the other is maximum.
Solution: Let one part be x. Then the other part is 84 - x
Letf (x) = x2 (84 - x) = 84x2 - x3
∴ f'(x) = `square`
and f''(x) = `square`
For extreme values, f'(x) = 0
∴ x = `square "or" square`
f(x) attains maximum at x = `square`
Hence, the two parts of 84 are 56 and 28.
Find the maximum and the minimum values of the function f(x) = x2ex.
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
When is a point in the domain of a function called a critical point?
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local maximum?
For a continuous function on a closed interval \[[a,b]\], which values must be compared to find absolute maxima and minima?
