Application of Derivatives

A point \(P\) is given on the circumference of a circle of radius \(r\). Chord \(QR\) is parallel to the tangent at \(P\). Determine the maximum possible area of the triangle \(PQR\).

Find all maxima and minima of the function
\($y = x{\left( {x - 1} \right)^2},0 \le x \le 2\)$
Also determine the area bounded by the curve \(y = x{\left( {x - 1} \right)^2}\),
the \(y\)-axis and the line \(y-2\).

Investigate for maxima and minimum the function
\($f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right]} dt\)$

The set of all \(x\) for which \(in\left( {1 + x} \right) \le x\) is equal to ..........

The smallest positive root of the equation, \(\tan x - x = 0\) lies in

Let \(f\) and \(g\) be increasing and decreasing functions, respectively from \(\left[ {0,\infty } \right)\) to \(\left[ {0,\infty } \right)\). Let \(h\left( x \right) = f\left( {g\left( x \right)} \right).\) If \(h\left( 0 \right) = 0,\) t...

Find the point on the curve \(\,\,\,4{x^2} + {a^2}{y^2} = 4{a^2},\,\,\,4 < {a^2} < 8\)
that is farthest from the point \((0, -2)\).

Let \(P\left( x \right) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + ...... + {a_n}{x^{2n}}\) be a polynomial in a real variable \(x\) with
\(0 < {a_0} < {a_1} < {a_2} < ..... < {a_n}.\) The function \(P(x)\) has

If the line \(ax+by+c=0\) is a normal to the curve \(xy=1\), then

Find all the tangents to the curve
\(y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,\) that are parallel to the line \(x+2y=0\).

Let \(f\left( x \right) = {\sin ^3}x + \lambda {\sin ^2}x, - {\pi \over 2} < x < {\pi \over 2}.\) Find the intervals in which \(\lambda\) should lie in order that \(f(x)\) has exactly one minimum and exactly one maximum.

For \(0 < a < x,\) the minimum value of the function \(lo{g_a}x + {\log _x}a\) is \(2\).

\(AB\) is a diameter of a circle and \(C\) is any point on the circumference of the circle. Then

The larger of \(\cos \left( {In\,\,\theta } \right)\) and \(In\) \(\left( {\cos \,\,\theta } \right)\) If \({e^{ - \pi /2}} < \theta < {\pi \over 2}\) is ..................

The normal to the curve \(\,x = a\left( {\cos \theta + \theta \sin \theta } \right)\), \(y = a\left( {\sin \theta - \theta \cos \theta } \right)\) at any point \('\theta '\) is such that

If \(a+b+c=0\), then the quadratic equation \(3a{x^2} + 2bx + c = 0\) has

If \(x-r\) is a factor of the polynomial \(f\left( x \right) = {a_n}{x^4} + ..... + {a_0},\) repeated \(m\) times \(\left( {1 < m \le n} \right)\), then \(r\) is a root of \(\left( x \right) = 0\) repeated \(m\) times.

The function \(y = 2{x^2} - In\,\left| x \right|\) is monotonically increasing for values of \(x\left( {x \ne 0} \right)\) satisfying the inequalities ......... and monotonically decreasing for values of \(x\) satisfying the inequalities .....

If \(y = a\,\,In\,x + b{x^2} + x\) has its extreamum values at \(x=-1\) and \(x=2\), then

Find the coordinates of the point on the curve \(y = {x \over {1 + {x^2}}}\)
where the tangent to the curve has the greatest slope.

Show that \(1+x\) \(In\left( {x + \sqrt {{x^2} + 1} } \right) \ge \sqrt {1 + {x^2}}\) for all \(x \ge 0\)

If \(f(x)\) and \(g(x)\) are differentiable function for \(0 \le x \le 1\) such that \(f(0)=2\), \(g(0)=0\), \(f(1)=6\); \(g(1)=2\), then show that there exist \(c\) satisfying \(0 < c < 1\) and \(f'(c)=2g'(c)\).

If \(a{x^2} + {b \over x} \ge c\) for all positive \(x\) where \(a>0\) and \(b>0\) show that \(27a{b^2} \ge 4{c^3}\).

Use the function \(f\left( x \right) = {x^{1/x}},x > 0\). to determine the bigger of the two numbers \({e^\pi }\) and \({\pi ^e}\)

Let \(x\) and \(y\) be two real variables such that \(x>0\) and \(xy=1\). Find the minimum value of \(x+y\).

For all \(x\) in \(\left[ {0,1} \right]\), let the second derivative \(f''(x)\) of a function \(f(x)\) exist and satisfy \(\left| {f''\left( x \right)} \right| < 1.\) If \(f(0)=f(1)\), then show that $$\left| {f\left( x \right)} \right| < 1...

Prove that the minimum value of \({{\left( {a + x} \right)\left( {b + x} \right)} \over {\left( {c + x} \right)}},\)
\(a,b > c,x > - c\) is \({\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}\).