Applications of Derivatives

The minimum distance of a point on the curve $y=x^2-4$ from the origin is :

Let $f(x)=\sin^4x+\cos^4x$. Then f is an increasing function in the interval :

Let C be a curve given by $y(x)=1+\sqrt{4x-3}$, $x>\dfrac34$. If P is a point on C, such that the tangent at P has slope $\dfrac23$, then a point through which the normal at P passes, is :

If A > 0, B > 0 and $A+B=\dfrac\pi6$, then the minimum value of tanA + tanB is :

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $V$ mm³, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container.

If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of $\dfrac{V}{250\pi}$ is

Let $y(x)$ be a solution of the differential equation $(1+e^x)y'+ye^x=1$. If $y(0)=2$, then which of the following statements is (are) true?

Let $f,g:[-1,2]\to\mathbb R$ 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:

	$x=-1$	$x=0$	$x=2$
$f(x)$	3	6	0
$g(x)$	0	1	−1

In each of the intervals $(-1,0)$ and $(0,2)$ the function $(f-3g)''$ never vanishes. Then the correct statement(s) is(are)

The correct statement(s) is(are)

The normal to the curve, $x^2+2xy-3y^2=0$, at (1, 1) :

Let $f(x)$ be a polynomial of degree four having extreme values at $x=1$ and $x=2$. If $\displaystyle\lim_{x\to0}\left[1+\frac{f(x)}{x^2}\right]=3$, then $f(2)$ is equal to :

The distance, from the origin, of the normal to the curve, $x=2\cos t+2t\sin t$, $y=2\sin t-2t\cos t$ at $t=\dfrac\pi4$, is :

If Rolle's theorem holds for the function $f(x)=2x^3+bx^2+cx$, $x\in[-1,1]$, at the point $x=\dfrac12$, then $2b+c$ equals :

Let the tangents drawn to the circle, $x^2+y^2=16$ from the point P(0, h) meet the $x$-axis at points A and B. If the area of ΔAPB is minimum, then h is equal to :

The equation of a normal to the curve, $\sin y=x\sin\left(\dfrac\pi3+y\right)$ at $x=0$, is :

Let k and K be the minimum and the maximum values of the function $f(x)=\dfrac{(1+x)^{0.6}}{1+x^{0.6}}$ in [0, 1] respectively, then the ordered pair (k, K) is equal to :

From the top of a 64 metres high tower, a stone is thrown upwards vertically with the velocity of 48 m/s. The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration $g=32$ m/s$^2$, is :

Let $f:(0,\infty)\to\mathbb R$ be given by
$$f(x)=\int_{1/x}^{x}e^{-\left(t+\frac1t\right)}\frac{dt}t.$$
Then

Let $a\in\mathbb R$ and let $f:\mathbb R\to\mathbb R$ be given by
$$f(x)=x^5-5x+a.$$
Then

Let $f:\left(-\frac\pi2,\frac\pi2\right)\to\mathbb R$ be given by
$$f(x)=\left(\log(\sec x+\tan x)\right)^3.$$
Then

The slope of the tangent to the curve $(y-x^5)^2=x(1+x^2)^2$ at the point $(1,3)$ is