Application of Derivatives

The real number \(k\) for which the equation, \(2{x^3} + 3x + k = 0\) has two distinct real roots in \(\left[ {0,\,1} \right]\)

A line is drawn through the point \((1, 2)\) to meet the coordinate axes at \(P\) and \(Q\) such that it forms a triangle \(OPQ,\) where \(O\) is the origin. If the area of the triangle \(OPQ\) is least, then the slope of the line \(PQ\) i...

Let \(a,b \in R\) be such that the function \(f\) given by \(f\left( x \right) = In\left| x \right| + b{x^2} + ax,\,x \ne 0\) has extreme values at \(x=-1\) and \(x=2\)
Statement-1 : \(f\) has local maximum at \(x=-1\) and at \(x=2\).
Stat...

A spherical balloon is filled with \(4500\pi\) cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of \(72\pi\) cubic meters per minute, then the rate (in meters per minute) at which the radius of the...

The shortest distance between line \(y-x=1\) and curve \(x = {y^2}\) is

For \(x \in \left( {0,{{5\pi } \over 2}} \right),\) define \(f\left( x \right) = \int\limits_0^x {\sqrt t \sin t\,dt.}\) Then \(f\) has

Let \(f:R \to R\) be a continuous function defined by
\($f\left( x \right) = {1 \over {{e^x} + 2{e^{ - x}}}}\)$
Statement - 1 : \(f\left( c \right) = {1 \over 3},\) for some \(c \in R\).
Statement - 2 : $$0 < f\left( x \right) \le {1 \over...

The equation of the tangent to the curve \(y = x + {4 \over {{x^2}}}\), that
is parallel to the \(x\)-axis, is

Let \(f:R \to R\) be defined by
\($f\left( x \right) = \left\{ {\matrix{ {k - 2x,\,\,if} & {x \le - 1} \cr {2x + 3,\,\,if} & {x > - 1} \cr } } \right.\)$
If \(f\)has a local minimum at \(x=-1\), then a possible value of \(k\)...

Given \(P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d\) such that \(x=0\) is the only
real root of \(P'\,\left( x \right) = 0.\) If \(P\left( { - 1} \right) < P\left( 1 \right),\) then in the interval \(\left[ { - 1,1} \right]:\)

How many real solutions does the equation
\({x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0\) have?

Suppose the cubic \({x^3} - px + q\) has three distinct real roots
where \(p>0\) and \(q>0\). Then which one of the following holds?

A value of \(c\) for which conclusion of Mean Value Theorem holds for the function \(f\left( x \right) = {\log _e}x\) on the interval \(\left[ {1,3} \right]\) is

The function \(f\left( x \right) = {\tan ^{ - 1}}\left( {\sin x + \cos x} \right)\) is an incresing function in

If \(p\) and \(q\) are positive real numbers such that \({p^2} + {q^2} = 1\), then the maximum value of \((p+q)\) is

Angle between the tangents to the curve \(y = {x^2} - 5x + 6\) at the points \((2,0)\) and \((3,0)\) is

The function \(f\left( x \right) = {x \over 2} + {2 \over x}\) has a local minimum at

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length \(x\). The maximum area enclosed by the park 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

A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2 \mathrm{~cm} / \mathrm{s}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20 \mathrm{~cm} / \ma...

A spherical iron ball \(10\) cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of \(50\) cm\(^3\) /min. When the thickness of ice is \(5\) cm, then the rate at which the thickness of ice decreases is

Let f be differentiable for all x. If f(1) = -2 and f'(x) \(\ge\) 2 for
x \(\in \left[ {1,6} \right]\), then

If the equation \({a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ........... + {a_1}x = 0\)
\({a_1} \ne 0,n \ge 2,\) has a positive root \(x = \alpha\), then the equation
$$n{a_n}{x^{n - 1}} + \left( {n - 1} \right){a_{n - 1}}{x^{n - 2}} + ..........

Area of the greatest rectangle that can be inscribed in the
ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\)

A function is matched below against an interval where it is supposed to be
increasing. Which of the following pairs is incorrectly matched?

If \(2a+3b+6c=0\), then at least one root of the equation
\(a{x^2} + bx + c = 0\) lies in the interval

A function \(y=f(x)\) has a second order derivative \(f''\left( x \right) = 6\left( {x - 1} \right).\) If its graph passes through the point \((2, 1)\) and at that point the tangent to the graph is \(y = 3x - 5\), then the function is :

The normal to the curve x = a(1 + cos \(\theta\)), \(y = a\sin \theta\) at \('\theta '\) always passes through the fixed point

A point on the parabola \({y^2} = 18x\) at which the ordinate increases at twice the rate of the abscissa is

If the function \(f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,\) where \(a>0,\) attains its maximum and minimum at \(p\) and \(q\) respectively such that \({p^2} = q\) , then \(a\) equals

The real number \(x\) when added to its inverse gives the minimum sum at \(x\) equal :

The maximum distance from origin of a point on the curve
\(x = a\sin t - b\sin \left( {{{at} \over b}} \right)\)
\(y = a\cos t - b\cos \left( {{{at} \over b}} \right),\) both \(a,b > 0\) is

If \(2a+3b+6c=0,\) \(\left( {a,b,c \in R} \right)\) then the quadratic equation \(a{x^2} + bx + c = 0\) has