Questions and Solutions
Question 1:
Find the maximum and minimum values, if any, of the following given by (i) f(x) = (2x - 1)^2 + 3 (ii) f(x) = 9x^2 + 12x + 2 (iii) f(x) = - (x - 1)^2 + 10 (iv) g(x) = x^3 + 1.
Question 2:
Find the maximum and minimum values, if any, of the following functions given by (i) f(x) = |x + 2| - 1 (ii) g(x) = - |x + 1| + 3 (iii) h(x) = sin(2x) + 5 (iv) f(x) = |sin 4x + 3| (v) h(x) = x + 4, x in (-1, 1).
Question 3:
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 (iii) h(x) = sin x + cos x, 0 < x < π/2 (iv) f(x) = sin x - cos x, 0 < x < 2π (v) f(x) = x^3 - 6x^2 + 9x + 15 (vi) g(x) = x/2 + 2/x, x > 0 (vii) g(x) = 1/(x^2 + 2) (viii) f(x) = x sqrt(1 - x), x > 0.
Question 4:
Prove that the following functions do not have maxima or minima: (i) f(x) = e^x (ii) g(x) = log x (iii) h(x) = x^3 + x^2 + x + 1.
Question 5:
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i) f(x) = x^3, x in [-2, 2] (ii) f(x) = sin x + cos x, x in [0, π] (iii) f(x) = 4x - 1/2 x^2, x in [-2, 9/2] (iv) f(x) = (x - 1)^2 + 3, x in [-3, 1].
Question 6:
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 - 24x - 18x^2.
Question 7:
Find the intervals in which the function f given by f(x) = x^3 + 1/x^3, x ≠ 0 is (i) Increasing (ii) Decreasing.
Question 8:
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Question 9:
What is the maximum value of the function sin x + cos x?
Question 10:
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].
Question 11:
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.
Question 12:
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Question 13:
Find two numbers whose sum is 24 and whose product is as large as possible.
Question 14:
Find two positive numbers x and y such that x + y = 60 and x y^3 is maximum.
Question 15:
Find two positive numbers x and y such that their sum is 35 and the product x^2 y^5 is maximum.
Question 16:
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Question 17:
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?
Question 18:
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 the maximum possible?
Question 19:
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Question 20:
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.
Question 21:
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.
Question 22:
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 circle is minimum?
Question 23:
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.
Question 24:
Show that the right circular cone of least curved surface and given volume has an altitude equal to √2 times the radius of the base.
Question 25:
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan^(-1)(√2).
Question 26:
Show that the semi-vertical angle of the right circular cone of given surface area and maximum volume is tan^(-1)(1/3).
Question 27:
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).
Question 28:
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.
Question 29:
The maximum value of [x(x + 1) + 1]^(1/3), 0 ≤ x ≤ 1 is: (A) (1/3)^(1/3) (B) 1/2 (C) 1 (D) 0.
Question 30:
For the function \(f(x)=x^{3}\), \(x=0\) is a point of: