Application of Derivatives

If \(a_{\alpha}\) is the greatest term in the sequence \(\alpha_{n}=\frac{n^{3}}{n^{4}+147}, n=1,2,3, \ldots\), then \(\alpha\) is equal to _____________.

The number of points, where the curve \(y=x^{5}-20 x^{3}+50 x+2\) crosses the \(\mathrm{x}\)-axis, is ____________.

Let a curve \(y=f(x), x \in(0, \infty)\) pass through the points \(P\left(1, \frac{3}{2}\right)\) and \(Q\left(a, \frac{1}{2}\right)\). If the tangent at any point \(R(b, f(b))\) to the given curve cuts the \(\mathrm{y}\)-axis at the point ...

A wire of length \(20 \mathrm{~m}\) is to be cut into two pieces. A piece of length \(l_{1}\) is bent to make a square of area \(A_{1}\) and the other piece of length \(l_{2}\) is made into a circle of area \(A_{2}\). If \(2 A_{1}+3 A_{2}\)...

The number of points on the curve \(y=54 x^{5}-135 x^{4}-70 x^{3}+180 x^{2}+210 x\) at which the normal lines are parallel to \(x+90 y+2=0\) is :

If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$
and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$ have a common extreme point, then $a+2 b+7$ is equal to :

If the equation of the normal to the curve \(y = {{x - a} \over {(x + b)(x - 2)}}\) at the point (1, \(-\)3) is \(x - 4y = 13\), then the value of \(a + b\) is equal to ___________.

Let \(f:(0,1)\to\mathbb{R}\) be a function defined \(f(x) = {1 \over {1 - {e^{ - x}}}}\), and \(g(x) = \left( {f( - x) - f(x)} \right)\). Consider two statements
(I) g is an increasing function in (0, 1)
(II) g is one-one in (0, 1)
Then,

Let \(x=2\) be a local minima of the function \(f(x)=2x^4-18x^2+8x+12,x\in(-4,4)\). If M is local maximum value of the function \(f\) in (\(-4,4)\), then M =

Let the function \(f(x) = 2{x^3} + (2p - 7){x^2} + 3(2p - 9)x - 6\) have a maxima for some value of \(x < 0\) and a minima for some value of \(x > 0\). Then, the set of all values of p is

The sum of the absolute maximum and minimum values of the function \(f(x)=\left|x^{2}-5 x+6\right|-3 x+2\) in the interval \([-1,3]\) is equal to :

Consider the triangles with vertices $A(2,1), B(0,0)$ and $C(t, 4), t \in[0,4]$.

If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6 \alpha+21 \beta$ is equal to ______...

\(\max _\limits{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=\)

If the local maximum value of the function \(f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right)\) , is \(\frac{k}{e}\), then \(\left(\frac{k}{e}\right)^{8}+\frac{k^{8}}{e^{5}}+k^{8}\) is equal to

Let \(f:[2,4] \rightarrow \mathbb{R}\) be a differentiable function such that \(\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)+f(x) \geq 1, x \in[2,4]\) with \(f(2)=\frac{1}{2}\) and \(f(4)=\frac{1}{4}\).
Consider th...

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm\(^2\)) is equal to :

The slope of tangent at any point (x, y) on a curve \(y=y(x)\) is \({{{x^2} + {y^2}} \over {2xy}},x > 0\). If \(y(2) = 0\), then a value of \(y(8)\) is :

Let the quadratic curve passing through the point \((-1,0)\) and touching the line \(y=x\) at \((1,1)\) be \(y=f(x)\). Then the \(x\)-intercept of the normal to the curve at the point \((\alpha, \alpha+1)\) in the first quadrant is ________...

Let \(\mathrm{g}(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x) > 0, x \in(0,1)\). If \(\mathrm{g}\) is decreasing in the interval \((0, a)\) and increasing in the interval \((\alpha, 1)\), then $$\tan ^{-1}(2 \alpha)+\tan ^{-1}\left(\frac{1}{...

A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infecte...

Let \(f(x) = 4{x^3} - 11{x^2} + 8x - 5,\,x \in R\). Then f :

If xy4 attains maximum value at the point (x, y) on the line passing through the points (50 + \(\alpha\), 0) and (0, 50 + \(\alpha\)), \(\alpha\) > 0, then (x, y) also lies on the line :

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square ...

Let f : R \(\to\) R be a function defined by f(x) = (x \(-\) 3)n1 (x \(-\) 5)n2, n1, n2 \(\in\) N. Then, which of the following is NOT true?

Let \(f(x)=3^{\left(x^{2}-2\right)^{3}+4}, x \in \mathrm{R}\). Then which of the following statements are true?
\(\mathrm{P}: x=0\) is a point of local minima of \(f\)
\(\mathrm{Q}: x=\sqrt{2}\) is a point of inflection of \(f\)
$$R: f^{\pr...

If the tangent to the curve \(y=x^{3}-x^{2}+x\) at the point \((a, b)\) is also tangent to the curve \(y = 5{x^2} + 2x - 25\) at the point (2, \(-\)1), then \(|2a + 9b|\) is equal to __________.

Let l be a line which is normal to the curve y = 2x2 + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to ___________.

The number of real solutions of \({x^7} + 5{x^3} + 3x + 1 = 0\) is equal to ____________.

If the minimum value of \(f(x)=\frac{5 x^{2}}{2}+\frac{\alpha}{x^{5}}, x>0\), is 14 , then the value of \(\alpha\) is equal to :

The function \(f(x)=x \mathrm{e}^{x(1-x)}, x \in \mathbb{R}\), is :

Let \(M\) and \(N\) be the number of points on the curve \(y^{5}-9 x y+2 x=0\), where the tangents to the curve are parallel to \(x\)-axis and \(y\)-axis, respectively. Then the value of \(M+N\) equals ___________.

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is \(\tan ^{-1} \frac{3}{4}\). Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square mete...

Let \(f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10\), \(x \in [ - 1,1]\). If [a, b] is the range of the function f, then 4a \(-\) b is equal to :

Let S be the set of all the natural numbers, for which the line \({x \over a} + {y \over b} = 2\) is a tangent to the curve \({\left( {{x \over a}} \right)^n} + {\left( {{y \over b}} \right)^n} = 2\) at the point (a, b), ab \(\ne\) 0. Then ...

The sum of the absolute minimum and the absolute maximum values of the function f(x) = |3x \(-\) x2 + 2| \(-\) x in the interval [\(-\)1, 2] is :

Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is :

Let the function \(f(x)=2 x^{2}-\log _{\mathrm{e}} x, x>0\), be decreasing in \((0, \mathrm{a})\) and increasing in \((\mathrm{a}, 4)\). A tangent to the parabola \(y^{2}=4 a x\) at a point \(\mathrm{P}\) on it passes through the point $$(8...

If the maximum value of \(a\), for which the function \(f_{a}(x)=\tan ^{-1} 2 x-3 a x+7\) is non-decreasing in \(\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)\), is \(\bar{a}\), then \(f_{\bar{a}}\left(\frac{\pi}{8}\right)\) is equal to :

Let \(f(x) = |(x - 1)({x^2} - 2x - 3)| + x - 3,\,x \in R\). If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ____________.

If the angle made by the tangent at the point (x0, y0) on the curve \(x = 12(t + \sin t\cos t)\), \(y = 12{(1 + \sin t)^2}\), \(0 < t < {\pi \over 2}\), with the positive x-axis is \({\pi \over 3}\), then y0 is equal to:

Water is being filled at the rate of 1 cm3 / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm2 / sec) at which the wet conical surface...

The curve \(y(x)=a x^{3}+b x^{2}+c x+5\) touches the \(x\)-axis at the point \(\mathrm{P}(-2,0)\) and cuts the \(y\)-axis at the point \(Q\), where \(y^{\prime}\) is equal to 3 . Then the local maximum value of \(y(x)\) is:

If the absolute maximum value of the function \(f(x)=\left(x^{2}-2 x+7\right) \mathrm{e}^{\left(4 x^{3}-12 x^{2}-180 x+31\right)}\) in the interval \([-3,0]\) is \(f(\alpha)\), then :

The sum of the maximum and minimum values of the function \(f(x)=|5 x-7|+\left[x^{2}+2 x\right]\) in the interval \(\left[\frac{5}{4}, 2\right]\), where \([t]\) is the greatest integer \(\leq t\), is ______________.

Let \(\lambda x - 2y = \mu\) be a tangent to the hyperbola \({a^2}{x^2} - {y^2} = {b^2}\). Then \({\left( {{\lambda \over a}} \right)^2} - {\left( {{\mu \over b}} \right)^2}\) is equal to :

The sum of absolute maximum and absolute minimum values of the function \(f(x) = |2{x^2} + 3x - 2| + \sin x\cos x\) in the interval [0, 1] is :

If the tangent at the point (x1, y1) on the curve \(y = {x^3} + 3{x^2} + 5\) passes through the origin, then (x1, y1) does NOT lie on the curve :

For the function \(f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1\), which one of the following is NOT correct?

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

Let \(\lambda\)\(^ *\) be the largest value of \(\lambda\) for which the function \({f_\lambda }(x) = 4\lambda {x^3} - 36\lambda {x^2} + 36x + 48\) is increasing for all x \(\in\) R. Then $${f_{{\lambda ^ * }}}(1) + {f_{{\lambda ^ * }}}( -...