- Prove that the minimum value of for , is .
- 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.
- 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.
- Determine the altitude of a cone with the greatest possible volume inscribed in a sphere of radius R.
- Find the sides of a rectangle with the greatest possible perimeter inscribed in a semicircle of radius R.
- Prove that in an ellipse, the distance between the centre and any normal does not exceed the difference between the semi-axes.
- Determine the altitude of a cylinder of the greatest possible volume which can be inscribed in a sphere of radius R.
- 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).
- 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 .
- 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 .
- Find the interval of monotonicity of the following functions: (i) (ii) (iii) (iv)
- Prove that if , then
- 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.
- Prove that for and for any positive a and b, the inequality is valid.
- Given that for all real x, and , prove that for all , and that for all .
- If for all , prove that at most once in .
- Suppose that a function f has a continuous second derivative, , , for all x. Show that for all x.
- Show that has exactly one root in .
- Find a polynomial such that . Prove that there is only one solution.
- Find a function, if possible whose domain is , , for all , , if and , if .
- Suppose that f is a continuous function on its domain and . Prove that f has at least one critical point in .
- 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.
- 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.
- 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.
- Let f be differentiable for all x, and suppose that , and that on and that on . Show that for all x.
- If b, c and d are constants, for what value of b will the curve have a point of inflection at ?
- Let and Find the critical points of g on
- 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.
- Let Find all possible real values of b such that has the smallest value at .
- 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.
- 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.
- Determine the points of maxima and minima of the function , with , where is a constant.
- 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.
- Let . Show that the equations has a unique root in the interval and identify it.
- Show that the following functions have at least one zero in the given interval: (i) , with (ii) with (iii) , with
- Show that all points of the curve at which the tangent is parallel to axis of x lie on a parabola.
- Show that the function f defined by , with has a maximum value with .
- Show that the function f defined by with has a minimum value at which and a maximum at when .
- If for all , then show that for all .
- Prove that , if .

Happy problem solving ! Practice makes man perfect.

Cheers,

Nalin Pithwa.