English

Write Sufficient Conditions for a Point X=C to Be a Point of Local Maximum. - Mathematics

Advertisements
Advertisements

Question

Write sufficient conditions for a point x = c to be a point of local maximum.

Short Note
Sum

Solution

We know that at the extreme points of a function f(x), the first order derivative of the function is equal to zero, i.e.
`f '(x) = 0 " at "x = c`

`⇒ f '(c) = 0`
Also, at the point of local maximum, the second order derivative of the function at the given point must be less than zero, i.e.
`f''(c) < 0`
shaalaa.com
  Is there an error in this question or solution?
Chapter 18: Maxima and Minima - Exercise 18.6 [Page 80]

APPEARS IN

RD Sharma Mathematics [English] Class 12
Chapter 18 Maxima and Minima
Exercise 18.6 | Q 2 | Page 80

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

f(x) = - (x-1)2+2 on R ?


f (x) = \[-\] | x + 1 | + 3 on R .


f(x) = x\[-\] 3x .


f(x) = x3  (x \[-\] 1).


`f(x)=2sinx-x, -pi/2<=x<=pi/2`


`f(x) = 2/x - 2/x^2,  x>0`


`f(x) = (x+1) (x+2)^(1/3), x>=-2` .


f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .


f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .


f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .


f(x) = (x \[-\] 1) (x \[-\] 2)2.


If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?


f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .


Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .


Determine two positive numbers whose sum is 15 and the sum of whose squares 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 lengths of the two pieces so that the combined area of the circle and the square is minimum?


Given the sum of the perimeters of a square and a circle, show that the sum of there areas is least when one side of the square is equal to diameter of the circle.


A tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?


Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r. 


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 ?


Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?


Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi  {cm}^3 .\]


An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.


The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?


Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]


Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\] 


Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .


Let f(x) = x3+3x\[-\] 9x+2. Then, f(x) has _________________ .


Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .


At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .


The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .


If x+y=8, then the maximum value of xy is ____________ .


If 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 ______________ .


f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .


The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .


The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .


Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×