Let = , ……….(i) = Now = 0 [Turning point] and it belongs to the given enclosed interval i.e., [0, 1]. At , from eq. (i), At from eq. (i), At , from eq. (i), Maximum value...
For all real values of x, the minimum value of 1-x+x^2/1+x+x^2 is (A) 0 (B) 1 (C) 3 (D) 1/ 3
Given: ……….(i) = Now = 0 and [Turning points] At , from eq. (i), At , from eq. (i), [Minimum value] Therefore, The correct option is option...
Choose the correct answer The point on the curve x^ 2 = 2y which is nearest to the point (0, 5) is (A) (2 2,4) (B) (2 2,0) (C) (0, 0) (D) (2, 2)
Equation of the curve is ……….(i) Let P be any point on the curve (i), then according to question, Distance between...
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin^−1(1/3) .
Let be the radius and be the height of the cone and semi-vertical angle be Total Surface area of cone (S) = (say) ……….(i) Volume of cone (V) = = = = [Using quotient rule]...
Show that the semi-vertical angle of the cone of the maximum value and of given slant height is tan ^−1√2
Let be the radius, be the height, be the slant height of given cone and be the semi-vertical angle of cone. ……….(i) Volume of the cone (V) = ……….(ii) V = = and Now ...
Show that the right circular cone of least curve surface and given volume has an altitude equal to time the radius of the base.
Let be the radius and be the height of the cone. Volume of the cone (V) = (say) ……….(i) And Surface area of the cone (S) = (say) ….(ii) = = and Now ……….(iii)...
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is of the volume of the sphere.
Let O be the centre and R be the radius of the given sphere, BM = and AM = In right angled triangle OMB, using Pythagoras theorem, OM2 + BM2 = OB2 ……….(i) Volume of a cone inscribed in...
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?
Let meters be the side of square and meters be the radius of the circle. Length of the wire = Perimeter of square + Circumference of circle = 28 ……….(i) Area of square = and Area of...
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimeters, find the dimensions of the can which has the minimum surface area.
Let be the radius of the circular base and be the height of closed right circular cylinder. According to the question, Volume of the cylinder …(i) Total surface area (S) = = = [From eq....
Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
Let be the radius of the circular base and be the height of closed right circular cylinder. Total surface area (S) = = (say) …..(i) Volume of cylinder = [From eq. (i)] and Now ...
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Let PQRS be the rectangle inscribed in a given circle with centre O and radius Let and be the length and breadth of the rectangle, i.e., and In right angled triangle PQR, using Pythagoras...
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 maximum ?
Dimensions of rectangular sheet are 45 cm and 24 cm. Let cm be the side of each of the four squares cut off from each corner. Then dimensions of the open box formed by folding the flaps after...
A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.
Each side of square piece of tin is 18 cm. Let cm be the side of each of the four squares cut off from each corner. Then dimensions of the open box formed by folding the flaps after cutting off...
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Let the two positive numbers are and ……….(i) Let [From eq. (i)] = and Now = 0 At is positive. is a point of local minima and is minimum when . Therefore, the...
Find two positive numbers x and y such that their sum is 35 and the product x ^2 y^ 5 is a maximum.
Given: ……….(i) Let [From eq. (i)] ……….(ii) Now or or or or Now is rejected because according to question, is a positive number. Also is rejected because from eq....
Find two positive numbers x and y such that x + y = 60 and xy^3 is maximum.
Given: ……….(i) Let P = [To be maximized] ……….(ii) Putting from eq. (i), in eq. (ii), P = …..(iii) Now It is clear that changes sign from positive to negative as increases...
Find two numbers whose sum is 24 and whose product is as large as possible
Let the two numbers be and According to the question, …….(i) And let is the product of and [From eq. (i)] and Now to find turning point, At , [Negative] is a point of...
Find the maximum and minimum values of x + sin 2x on [0, 2π]
Let Now = = = where Z For But , therefore For = and For But , therefore Therefore, it is clear that the only turning point of given by which belong to given closed...
It is given that at x = 1, the function x^ 4 – 62x^ 2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.
Let Since, attains its maximum value at in the interval [0, 2], therefore
Find the maximum value of 2x ^3 – 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [–3, –1].
Let Now or [Turning points] For Interval [1, 3], is turning point. At At At Therefore, maximum value of is 89. For Interval is turning point. At At At Therefore,...
What is the maximum value of the function sin x + cos x?
Let Now = [Turning point] = = = = If is even, then If is odd, then Therefore, maximum value of is and minimum value...
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Let Now Putting Now = = = Putting Also and Since attains its maximum value 1 at and Therefore, the required points...
Find both the maximum value and the minimum value of 3x ^4 – 8x^ 3 + 12x ^2 – 48x + 25 on the interval [0, 3].
Let on [0, 3] Now or Since is imaginary, therefore it is rejected. is turning point. At = At At Therefore, absolute minimum value is and absolute maximum value is...
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 – 72x – 18x^ 2
Profit function and Now At , [Negative] has a local maximum value at . At , Maximum profit = = 41 + 16 – 8 = 49
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (iii) f (x) = 4x-1/2x^2, x∈[-2,9/2] (iv) f(x)=(x-1)^2+3, x∈[-3,1]
(iii) Given: Now At At At Therefore, absolute minimum value is and absolute maximum value is 8. (iv) Given: Now At At Therefore, absolute minimum value is and absolute...
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i) f(x) = x^ 3 , x ∈ [– 2, 2] (ii) f (x) = sin x + cos x , x ∈ [0, π]
(i) Given: Now At At At Therefore, absolute minimum value of is and absolute maximum value is 8. (ii) Given: Now [Positive] is in I quadrant. [ ] Therefore,...
Prove that the following functions do not have maxima or minima:(iii) h (x) = x^ 3 + x^ 2 + x +1
(iii) Given: Now = = Here, values of are imaginary. does not have maxima or minima.
Prove that the following functions do not have maxima or minima: (i) f(x) = e^ x (ii) g(x) = log x
(i) Given: Now But this gives no real value of Therefore, there is no turning point. does not have maxima or minima. (ii) Given: Now 1 = 0 But this is not possible. Therefore,...
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:(vii) g(x)=1/(x^2+2) (viii) f (x )=x√1- x , 0 <x<1
(vii) Given: and = Now [Turning point] At [Negative] is a point of local maxima and local maximum value is (viii) Given: = = = And = = Now = 0 is a point of...
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:(v) f(x) = x ^3 – 6x^ 2 + 9x + 15 (vi) x/2+2 /x , x>0
(v) Given: and Now or [Turning points] At [Negative] is a point of local maxima and local maximum value is At [Positive] is a point of local minima and local minimum...
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: (iii) h(x) = sin x + cos x, 0 2 x π < < (iv) f(x) = sin x – cos x, 0 2 < < π
(iii) Given: ……….(i) and Now [Positive] can have values in both I and III quadrant. But, therefore, is only in I quadrant. = [Turning point] At = ...
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: (i) f(x) = x ^2 (ii) g(x) = x^ 3 – 3x
(i) Given: and Now [Turning point] Again, when , [Positive] Therefore, is a point of local minima and local minimum value = (ii) Given: and Now or [Turning...
Find the maximum and minimum values, if any, of the following functions given by (v) h(x) = x + 1, x ∈ (– 1, 1)
(v) Given: ……….(i) Since Adding 1 to both sides, Therefore, neither minimum value not maximum value of exists.
Find the maximum and minimum values, if any, of the following functions given by (iii) h(x) = sin(2x) + 5 (iv) f(x) = |sin 4x + 3|
(iii) Given: ……….(i) Since for all R Adding 5 to all sides, Therefore, minimum value of is 4 and maximum value is 6. (iv) Given: Since for all R Adding 3 to all sides, ...
Find the maximum and minimum values, if any, of the following functions given by (iii) f(x) = – (x – 1)^2 + 10 (iv) g(x) = x ^3 + 1
(iii) Given: ……….(i) Since for all R Multiplying both sides by and adding 10 both sides, [Using eq. (i)] Therefore, maximum value of is 10 which is obtained when i.e., And therefore,...
Find the maximum and minimum values, if any, of the following functions given by (i) f (x) = (2x – 1)^2 + 3 (ii) f(x) = 9x ^2 + 12x + 2
(i) Given: Since for all R Adding 3 both sides, Therefore, the minimum value of is 3 when , i.e., This function does not have a maximum value. (ii) Given: = ……….(i)...