Mathematics Hothouse

1. Prove that the minimum value of for , is .
2. A cylindrical vessel of volume cubic meters, open at the top, is to be manufactured from a sheet of metal. Find the dimensions of the vessel so that the amount of metal used is the least possible.
3. Assuming that the petrol burnt in driving a motor boat varies as the cube of its velocity, show that the most economical speed when going against a current of c kmph is kmph.
4. Determine the altitude of a cone with the greatest possible volume inscribed in a sphere of radius R.
5. Find the sides of a rectangle with the greatest possible perimeter inscribed in a semicircle of radius R.
6. Prove that in an ellipse, the distance between the centre and any normal does not exceed the difference between the semi-axes.
7. Determine the altitude of a cylinder of the greatest possible volume which can be inscribed in a sphere of radius R.
8. Break up the number 8 into two parts such that the sum of their cubes is the least possible. (Use calculus techniques, not intelligent guessing).
9. Find the greatest and least possible values of the following functions on the given interval: (i) on . (ii) on (iii) on with and (iv) on (iv) on .
10. Prove the following inequalities: (in these subset of questions, you can try to use pure algebraic methods, apart from calculus techniques to derive alternate solutions): (i) for (ii) (iii) for . (iv) for (iv) for .
11. Find the interval of monotonicity of the following functions: (i) (ii) (iii) (iv)
12. Prove that if , then
13. On the graph of the function where , find the point such that the segment of the tangent to the graph of the function at the point intercepted between the point M and the y-axis, is the shortest.
14. Prove that for and for any positive a and b, the inequality is valid.
15. Given that for all real x, and , prove that for all , and that for all .
16. If for all , prove that at most once in .
17. Suppose that a function f has a continuous second derivative, , , for all x. Show that for all x.
18. Show that has exactly one root in .
19. Find a polynomial such that . Prove that there is only one solution.
20. Find a function, if possible whose domain is , , for all , , if and , if .
21. Suppose that f is a continuous function on its domain and . Prove that f has at least one critical point in .
22. A right circular cone is inscribed in a sphere of radius R. Find the dimensions of the cone, if its volume is to be maximum.
23. Estimate the change in volume of right circular cylinder of radius R and height h when the radius changes from to and the height does not change.
24. For what values of a, m and b does the function: , when ; , when ; and , when satisfy the hypothesis of the Lagrange’s Mean Value Theorem.
25. Let f be differentiable for all x, and suppose that , and that on and that on . Show that for all x.
26. If b, c and d are constants, for what value of b will the curve have a point of inflection at ?
27. Let and Find the critical points of g on
28. Find a point P on the curve so that the area of the triangle formed by the tangent at P and the co-ordinate axes is minimum.
29. Let Find all possible real values of b such that has the smallest value at .
30. The circle cuts the x-axis at P and Q. Another circle with centre Q and variable radius intersects the first circle at R above the x-axis and line segment PQ at x. Find the maximum area of the triangle PQR.
31. A straight line L with negative slope passes through the point and cuts the positive co-ordinate axes at points P and Q. Find the absolute minimum value of , as L varies, where O is the origin.
32. Determine the points of maxima and minima of the function , with , where is a constant.
33. Let be a fixed point, where . A straight line passing through this point cuts the positive direction of the co-ordinate axes at points P and Q. Find the minimum area of the triangle , O being the origin.
34. Let . Show that the equations has a unique root in the interval and identify it.
35. Show that the following functions have at least one zero in the given interval: (i) , with (ii) with (iii) , with
36. Show that all points of the curve at which the tangent is parallel to axis of x lie on a parabola.
37. Show that the function f defined by , with has a maximum value with .
38. Show that the function f defined by with has a minimum value at which and a maximum at when .
39. If for all , then show that for all .
40. Prove that , if .

Happy problem solving ! Practice makes man perfect.

Cheers,

Nalin Pithwa.