Application of Derivatives

Prove that for \(x \in \left[ {0,{\pi \over 2}} \right],\) \(\sin x + 2x \ge {{3x\left( {x + 1} \right)} \over \pi }\). Explain
the identity if any used in the proof.

Using Rolle's theorem, prove that there is at least one root
in \(\left( {{{45}^{1/100}},46} \right)\) of the polynomial
\(P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035\).

In \(\left[ {0,1} \right]\) Languages Mean Value theorem is NOT applicable to

Tangent is drawn to ellipse
\({{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \in \left( {0,\pi /2} \right)} \right)\).
Then the value of \(\theta\) such that sum of in...

If \(P(1)=0\) and \({{dp\left( x \right)} \over {dx}} > P\left( x \right)\) for all \(x \ge 1\) then prove that
\(P(x)>0\) for all \(x>1\).

Find a point on the curve \({x^2} + 2{y^2} = 6\) whose distance from
the line \(x+y=7\), is minimum.

If the function \(f:\left[ {0,4} \right] \to R\) is differentiable then show that
(i)\(\,\,\,\,\,\) For \(a, b\)\(\,\,\)$$ \in \left( {0,4} \right),{\left( {f\left( 4 \right)} \right)^2} - {\left( {f\left( 0 \right)} \right)^2} = gf'\left(...

Using the relation \(2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0\) or otherwise,
prove that \(\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]\)

The length of a longest interval in which the function \(3\,\sin x - 4{\sin ^3}x\) is increasing, is

The point(s) in the curve \({y^3} + 3{x^2} = 12y\) where the tangent is vertical, is (are)

If \(f\left( x \right) = x{e^{x\left( {1 - x} \right)}},\) then \(f(x)\) is

Let \(f\left( x \right) = \left( {1 + {b^2}} \right){x^2} + 2bx + 1\) and let \(m(b)\) be the minimum value of \(f(x)\). As \(b\) varies, the range of \(m(b)\) is

The triangle formed by the tangent to the curve \(f\left( x \right) = {x^2} + bx - b\) at the point \((1, 1)\) and the coordinate axex, lies in the first quadrant. If its area is \(2\), then the value of \(b\) is

Let \(- 1 \le p \le 1\). Show that the equation \(4{x^3} - 3x - p = 0\)
has a unique root in the interval \(\left[ {1/2,\,1} \right]\) and identify it.

Consider the following statements in \(S\) and \(R\)
\(S:\) \(\,\,\,\)$ Both \(\sin \,\,x\) and \(\cos \,\,x\) are decreasing functions in the interval \(\left( {{\pi \over 2},\pi } \right)\)
\(R:\)\(\,\,\,\) If a differentiable function ...

For all \(x \in \left( {0,1} \right)\)

Let \(f\left( x \right) = \left\{ {\matrix{ {\left| x \right|,} & {for} & {0 < \left| x \right| \le 2} \cr {1,} & {for} & {x = 0} \cr } } \right.\) then at \(x=0\), \(f\) has

If the normal to the curve \(y = f\left( x \right)\) and the point \((3, 4)\) makes an angle \({{{3\pi } \over 4}}\) with the positive \(x\)-axis, then \(f'\left( 3 \right) =\)

Let \(f\left( x \right) = \int {{e^x}\left( {x - 1} \right)\left( {x - 2} \right)dx.}\) Then \(f\) decreases in the interval

Suppose \(p\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + .......... + {a_n}{x^n}.\) If
\(\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|\) for all \(x \ge 0\), prove that
$$\left| {{a_1} + 2{a_2} + ........ + n...

The function \(f(x)=\) \({\sin ^4}x + {\cos ^4}x\) increases if

The function \(f\left( x \right) = \int\limits_{ - 1}^x {t\left( {{e^t} - 1} \right)\left( {t - 1} \right){{\left( {t - 2} \right)}^3}\,\,\,{{\left( {t - 3} \right)}^5}}\) \(dt\) has a local minimum at \(x=\)

The number of values of \(x\) where the function
\(f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)\) attains its maximum is

A curve \(C\) has the property that if the tangent drawn at any point \(P\) on \(C\) meets the co-ordinate axes at \(A\) and \(B\), then \(P\) is the mid-point of \(AB\). The curve passes through the point \((1, 1)\). Determine the equation...

Let \(h\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \right)^3}\) for every real number \(x\). Then

If \(f\left( x \right) = {{{x^2} - 1} \over {{x^2} + 1}},\) for every real number \(x\), then the minimum value of \(f\)

Suppose \(f(x)\) is a function satisfying the following conditions
(a) \(f(0)=2,f(1)=1\),
(b) \(f\)has a minimum value at \(x=5/2\), and
(c) for all \(x\),
$$$f'\left( x \right) = \matrix{
{2ax} & {2ax - 1} & {2ax + b + 1} \cr
b ...

Let \(a+b=4\), where \(a<2,\) and let \(g(x)\) be a differentiable function.
If \({{dg} \over {dx}} > 0\) for all \(x\), prove that \(\int_0^a {g\left( x \right)dx + \int_0^b {g\left( x \right)dx} }\)
increases as \((b-a)\) increases.

If \(f\left( x \right) = {x \over {\sin x}}\) and \(g\left( x \right) = {x \over {\tan x}}\), where \(0 < x \le 1\), then in this interval

Determine the points of maxima and minima of the function
\(f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,\) where \(b \ge 0\) is a constant.

A curve \(y=f(x)\) passes through the point \(P(1, 1)\). The normal to the curve at \(P\) is \(a(y-1)+(x-1)=0\). If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of t...

Let \(f\left( x \right) = \left\{ {\matrix{ {x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr {x + a{x^2} - {x^3},\,x > 0} \cr } } \right.\)
Where a is a positive constant. Find the interval in which \(f'(x)\) is increasing.

On the interval \(\left[ {0,1} \right]\) the function \({x^{25}}{\left( {1 - x} \right)^{75}}\) takes its maximum value at the point

The function \(f\left( x \right) = {{in\,\left( {\pi + x} \right)} \over {in\,\left( {e + x} \right)}}\) is

The slope of the tangent to a curve \(y = f\left( x \right)\) at \(\left[ {x,\,f\left( x \right)} \right]\) is \(2x+1\). If the curve passes through the point \(\left( {1,2} \right)\), then the area bounded by the curve, the \(x\)-axis and ...

Let \((h, k)\) be a fixed point, where \(h > 0,k > 0.\). A straight line passing through this point cuts the possitive direction of the coordinate axes at the points \(P\) and \(Q\). Find the minimum area of the triangle \(OPQ\), \(O\) bein...

Let \(C\) be the curve \({y^3} - 3xy + 2 = 0\). If \(H\) is the set of points on the curve \(C\) where the tangent is horizontal and \(V\) is the set of the point on the curve \(C\) where the tangent is vertical then \(H=\).............. an...

The curve \(y = a{x^3} + b{x^2} + cx + 5\), touches the \(x\)-axis at \(P(-2, 0)\) and cuts the \(y\) axis at a point \(Q\), where its gradient is \(3\). Find \(a, b, c\).

Which one of the following curves cut the parabola \({y^2} = 4ax\) at right angles?

Let \(P\) be a variable point on the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) with foci \({F_1}\) and \({F_2}\). If \(A\) is the area of the triangle \(P{F_1}{F_2}\) then the maximum value of \(A\) is ..........

The function defined by \(f\left( x \right) = \left( {x + 2} \right){e^{ - x}}\)

The circle \({x^2} + {y^2} = 1\) cuts the \(x\)-axis at \(P\) and \(Q\). Another circle with centre at \(Q\) and variable radius intersects the first circle at \(R\) above the \(x\)-axis and the line segment \(PQ\) at \(S\). Find the maximu...

If \(f\left( x \right) = \left\{ {\matrix{ {3{x^2} + 12x - 1,} & { - 1 \le x \le 2} \cr {37 - x} & {2 < x \le 3} \cr } } \right.\) then:

Find the equation of the normal to the curve
\(y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)\) at \(x=0\)

Let \(f\left( x \right) = \left\{ {\matrix{ { - {x^3} + {{\left( {{b^3} - {b^2} + b - 1} \right)} \over {\left( {{b^2} + 3b + 2} \right)}},} & {0 \le x < 1} \cr {2x - 3} & {1 \le x \le 3} \cr } } \right.\)
Find all possible re...

In this questions there are entries in columns \(I\) and \(II\). Each entry in column \(I\) is related to exactly one entry in column \(II\). Write the correct letter from column \(II\) against the entry number in column \(I\) in your answe...

A cubic \(f(x)\) vanishes at \(x=2\) and has relative minimum / maximum at \(x=-1\) and \(x = {1 \over 3}\) if \(\int\limits_{ - 1}^1 {f\,\,dx = {{14} \over 3}}\), find the cubic \(f(x)\).

What normal to the curve \(y = {x^2}\) forms the shortest chord?

A window of perimeter \(P\) (including the base of the arch) is in the form of a rectangle surmounded by a semi circle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass transmits ...

Show that \(2\sin x + \tan x \ge 3x\) where \(0 \le x < {\pi \over 2}\).