- 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.