मराठी
महाराष्ट्र राज्य शिक्षण मंडळएचएससी वाणिज्य (इंग्रजी माध्यम) इयत्ता १२ वी

State whether the following statement is True or False: An absolute maximum must occur at a critical point or at an end point.

Advertisements
Advertisements

प्रश्न

State whether the following statement is True or False:

An absolute maximum must occur at a critical point or at an end point.

पर्याय

  • True

  • False

MCQ
चूक किंवा बरोबर
Advertisements

उत्तर

This statement is True.

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 4: Applications of Derivatives - Miscellaneous Exercise 4 [पृष्ठ ११४]

संबंधित प्रश्‍न

Find the maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 1.


Find the maximum and minimum value, if any, of the following function given by g(x) = − |x + 1| + 3.


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


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:

g(x) = x3 − 3x


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:

`g(x) = x/2 + 2/x, x > 0`


Find the absolute maximum value and the absolute minimum value of the following function in the given interval:

f (x) = (x −1)2 + 3, x ∈[−3, 1]


It is given that at x = 1, the function x4− 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.


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 altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3.`


A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.


Solve the following :  A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.


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.


Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.


If x + y = 3 show that the maximum value of x2y is 4.


Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`


If y = x3 + x2 + x + 1, then y ____________.


Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].


Read the following passage and answer the questions given below.


The temperature of a person during an intestinal illness is given by f(x) = 0.1x2 + mx + 98.6, 0 ≤ x ≤ 12, m being a constant, where f(x) is the temperature in °F at x days.

  1. Is the function differentiable in the interval (0, 12)? Justify your answer.
  2. If 6 is the critical point of the function, then find the value of the constant m.
  3. Find the intervals in which the function is strictly increasing/strictly decreasing.
    OR
    Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.

If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.


Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b is ______.


The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.


The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.


The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` 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.


Which statement gives the meaning of a local maximum at \[c\]?


Let \[c\] be a critical point of a continuous function \[f\]. If \[f'(x)\] changes sign from positive to negative as \[x\] passes through \[c\], what is \[c\]?


What are the critical points obtained from \[12x(x-1)(x+2)=0\]?


For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=1\]?


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×