Application of Derivatives

Consider the function $f : (0, \infty) \to (-\infty, \infty)$ given by
$f(x) = \sqrt{x} \log_e(x) - x + 1$.Then which one of the following statements is TRUE?

Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be defined by
$f(x) = \begin{cases} \dfrac{6x + \sin x}{2x + \sin x}, & \text{if } x \neq 0, \\ \dfrac{7}{3}, & \text{if } x = 0. \end{cases}$
Then which of the following statements is ...

Let $Q$ be the cube with the set of vertices $\left\{\left(x_1, x_2, x_3\right) \in \mathbb{R}^3: x_1, x_2, x_3 \in\{0,1\}\right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ b...

Let

\(\alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)}\)

Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by

\(g(x)=2^{\alpha x}+2^{\alpha(1-x)} .\)

Then, which of the following stateme...

Consider the rectangles lying the region \(\left\{ {(x,y) \in R \times R:0\, \le \,x\, \le \,{\pi \over 2}} \right.\) and \(\left. {0\, \le \,y\, \le \,2\sin (2x)} \right\}\)and having one side on the X-axis. The area of the rectangle whic...

Let, \(f(x) = {{\sin \pi x} \over {{x^2}}}\), x > 0Let x1 < x2 < x3 < ... < xn < ... be all the points of local maximum of f and y1 < y2 < y3 < ... < yn < ... be all the points of local minimum of f.Then which of the following options is/ar...

Let f : R \(\to\) R be given by\(f(x) = (x - 1)(x - 2)(x - 5)\). Define\(F(x) = \int\limits_0^x {f(t)dt}\), x > 0Then which of the following options is/are correct?

For each positive integer n, let \({y_n} = {1 \over n}(n + 1)(n + 2)...{(n + n)^{{1 \over n}}}\). For x\(\in\)R, let [x] be the greatest integer less than or equal to x. If \(\mathop {\lim }\limits_{n \to \infty } {y_n} = L\), then the va...

If \(f(x) = \left| {\matrix{ {\cos 2x} & {\cos 2x} & {\sin 2x} \cr { - \cos x} & {\cos x} & { - \sin x} \cr {\sin x} & {\sin x} & {\cos x} \cr } } \right|\),then

f : R \(\to\) R is a differentiable function such that f'(x) > 2f(x) for all x\(\in\)R, and f(0) = 1 then

Which of the following options is the only CORRECT combination?

Which of the following options is the only INCORRECT combination?

Which of the following options is the only CORRECT combination?

Let f: R \(\to \left( {0,\infty } \right)\) and g : R \(\to\) R be twice differentiable functions such that f'' and g'' are continuous functions on R. Suppose f'\((2)\) \(=\) g\((2)=0\), f''\((2)\)\(\ne 0\) and g'\((2)\) \(\ne 0\). If ...

The least value of a \(\in R\) for which \(4a{x^2} + {1 \over x} \ge 1,\), for all \(x>0\). is

Let \(f, g :\) \(\left[ { - 1,2} \right] \to R\) be continuous functions which are twice differentiable on the interval \((-1, 2)\). Let the values of f and g at the points \(-1, 0\) and \(2\) be as given in the following table:

.tg {bord...

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of \(V\) \(m{m^3}\), has a \(2\) mm thick solid wall and is open at the top. The bottom of the container is a soli...

The slope of the tangent to the curve \({\left( {y - {x^5}} \right)^2} = x{\left( {1 + {x^2}} \right)^2}\) at the point \((1, 3)\) is

Let \(f:\left[ {0,1} \right] \to R\) (the set of all real numbers) be a function. Suppose the function \(f\) is twice differentiable, \(f(0) = f(1)=0\) and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x}...

Let \(f:\left[ {0,1} \right] \to R\) (the set of all real numbers) be a function. Suppose the function \(f\) is twice differentiable, \(f(0) = f(1)=0\) and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x}...

The function \(f(x) = 2\left| x \right| + \left| {x + 2} \right| - \left| {\left| {x + 2} \right| - 2\left| x \right|} \right|\) has a local minimum or a local maximum at x =

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio \(8:15\) is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squar...

Let \(f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}\) for all \(x \in IR\) and let
\(g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt}\) ...

If \(f\left( x \right) = \int_0^x {{e^{{t^2}}}} \left( {t - 2} \right)\left( {t - 3} \right)dt\) for all \(x \in \left( {0,\infty } \right),\) then

Let \(f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}\) for all \(x \in IR\) and let
\(g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt}\) ...

Let \(f:IR \to IR\) be defined as \(f\left( x \right) = \left| x \right| + \left| {{x^2} - 1} \right|.\) The total number of points at which \(f\) attains either a local maximum or a local minimum is

Let \(p(x)\) be a real polynomial of least degree which has a local maximum at \(x=1\) and a local minimum at \(x=3\). If \(p(1)=6\) and \(p(3)=2\), then \(p'(0)\) is

Let \(f\) be a function defined on \(R\) (the set of all real numbers)
such that \(f'\left( x \right) = 2010\left( {x - 2009} \right){\left( {x - 2010} \right)^2}{\left( {x - 2011} \right)^3}{\left( {x - 2012} \right)^4}\) for all $$x \in...

Let \(f\) be a real-valued differentiable function on \(R\) (the set of all real numbers) such that \(f(1)=1\). If the \(y\)-intercept of the tangent at any point \(P(x,y)\) on the curve \(y=f(x)\) is equal to the cube of the abscissa of $$...

The maximum value of the function \(f(x) = 2{x^3} - 15{x^2} + 36x - 48\) on the set \(A = \{ x|{x^2} + 20 \le 9x|\}\) is __________.

For the function
\($f\left( x \right) = x\cos \,{1 \over x},x \ge 1,\)$

Let \(p(x)\) be a polynomial of degree \(4\) having extremum at
\(x = 1,2\) and \(\mathop {\lim }\limits_{x \to 0} \left( {1 + {{p\left( x \right)} \over {{x^2}}}} \right) = 2\).
Then the value of \(p (2)\) is

The total number of local maxima and local minima of the function \(f(x) = \left\{ {\matrix{ {{{(2 + x)}^3},} & { - 3 < x \le - 1} \cr {{x^{2/3}},} & { - 1 < x < 2} \cr } } \right.\) is

Let \(f(x)\) be differentiable on the interval (0, \(\infty\)) such that \(f(1)=1\), and \(\mathop {\lim }\limits_{t \to x} {{{t^2}f(x) - {x^2}f(t)} \over {t - x}} = 1\) for each \(x > 0\). Then \(f(x)\) is

If a continuous function \(f\) defined on the real line \(R\), assumes positive and negative values in \(R\) then the equation \(f(x)=0\) has a root in \(R\). For example, if it is known that a continuous function \(f\) on \(R\) is positive...

The tangent to the curve \(y = {e^x}\) drawn at the point \(\left( {c,{e^c}} \right)\) intersects the line joining the points \(\left( {c - 1,{e^{c - 1}}} \right)\) and \(\left( {c + 1,{e^{c + 1}}} \right)\)

If a continuous function \(f\) defined on the real line \(R\), assumes positive and negative values in \(R\) then the equation \(f(x)=0\) has a root in \(R\). For example, if it is known that a continuous function \(f\) on \(R\) is positive...

If a continuous function \(f\) defined on the real line \(R\), assumes positive and negative values in \(R\) then the equation \(f(x)=0\) has a root in \(R\). For example, if it is known that a continuous function \(f\) on \(R\) is positive...

If $f(x)$ is a twice differentiable function such that $f(A)=0, f(B)=2, f(C)=-1, f(D)=2$, $f(e)=0$, where $a < b < c < d < e$, then the minimum number of zeroes of $g(x)=\left(f^{\prime}(x)\right)^2 +f^{\prime \prime}(x) f(x)$ in the interv...

$$ \begin{aligned} & f(x)=\left\{\begin{array}{cc} e^x, & 0 \leq x \leq 1 \\ 2-e^{x-1}, & 1 < x \leq 2 \\ x-e, & 2 < x \leq 3 \end{array} \quad\right. \text { and } \\ & g(x)=\int_0^x f(t) d t, x \in[1,3] \text { then } g(x) \text { has } \...

$f(x)$ is cubic polynomial which has local maximum at $x=-1$. If $f(2)=18, f(1)=-1$ and $f(x)$ has local minima at $x=0$, then

A tangent drawn to the curve $y=f(x)$ at $\mathrm{P}(x, y)$ cuts the X -axis and Y -axis at A and B respectively such that $\mathrm{BP}: \mathrm{AP}=3: 1$, given that $f(1)=1$, then

If \(f(x)\) is a twice differentiable function such that \(f(A)=0, f(B)=2, f(C)=-1, f(D)=2\), \(f(e)=0\), where \(a < b < c < d < e\), then the minimum number of zeroes of \(g(x)=\left(f'(x)\right)^{2}+f''(x) f(x)\) in the interval $$[a, e]...

If \(P(x)\) is a polynomial of degree less than or equal to \(2\) and \(S\) is the set of all such polynomials so that \(P(0)=0\), \(P(1)=1\) and \(P'\left( x \right) > 0\,\,\forall x \in \left[ {0,1} \right],\) then

If \(p(x)\) be a polynomial of degree 3 satisfying \(p(-1)=10, p(1)=-6\) and \(p(x)\) has maximum at \(x=-1\) and \(p'(x)\) has minima at \(x=1\). Find the distance between the local maximum and local minimum of the curve.

If \(\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right| \leq\left(x_{1}-x_{2}\right)^{2}\), for all \(x_{1}, x_{2} \in\) \(\mathbb{R}\). Find the equation of tangent to the curve \(y=f(x)\) at the point \((1,2)\).

If \(p(x)\) be a polynomial of degree \(3\) satisfying \(p(-1)=10, p(1)=-6\) and \(p(x)\) has maxima at \(x=-1\) and \(p'(x)\) has minima at \(x=1\). Find the distance between the local maxima and local minima of the curve.

If \(\left| {f\left( {{x_1}} \right) - f\left( {{x_2}} \right)} \right| < {\left( {{x_1} - {x_2}} \right)^2},\) for all \({x_1},{x_2} \in R\). Find the equation of tangent to the cuve \(y = f\left( x \right)\) at the point \((1, 2)\).

If \(f\left( x \right) = {x^3} + b{x^2} + cx + d\) and \(0 < {b^2} < c,\) then in \(\left( { - \infty ,\infty } \right)\)

If \(f\left( x \right) = {x^a}\log x\) and \(f\left( 0 \right) = 0,\) then the value of \(\alpha\) for which Rolle's theorem can be applied in \(\left[ {0,1} \right]\) is