Applications of Derivatives

For the function
$$f(x)=4\log_e(x-1)-2x^2+4x+5,\quad x>1,$$
which one of the following is NOT correct ?

If the tangent at the point $(x_1,y_1)$ on the curve $y=x^3+3x^2+5$ passes through the origin, then $(x_1,y_1)$ does NOT lie on the curve :

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

Let $\displaystyle\max_{0\le x\le2}\left\{\frac{9-x^2}{5-x}\right\}=\alpha$ and $\displaystyle\min_{0\le x\le2}\left\{\frac{9-x^2}{5-x}\right\}=\beta$.

If $\displaystyle\int_{\beta-\frac83}^{2\alpha-1}\max\left\{\frac{9-x^2}{5-x},x\right\}dx=\alpha_1+\alpha_2\log_e\left(\frac8{15}\right)$ then $\alpha_1+\alpha_2$ is equal to ______.

A particle is moving in the $xy$-plane along a curve $C$ passing through the point $(3,3)$. The tangent to the curve $C$ at the point $P$ meets the $x$-axis at $Q$. If the $y$-axis bisects the segment $PQ$, then $C$ is a parabola with

Let the maximum area of the triangle that can be inscribed in the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}4=1$, $a>2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $y$-axis, be $6\sqrt3$. Then the eccentricity of the ellipse is:

The number of distinct real roots of the equation $x^7-7x-2=0$ is

The slope of normal at any point $(x,y)$, $x>0$, $y>0$ on the curve $y=y(x)$ is given by $\dfrac{x^2}{xy-x^2y^2-1}$. If the curve passes through the point $(1,1)$, then $e\cdot y(e)$ is equal to

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 $x\in\mathbb R$. Then $f_{\lambda^*}(1)+f_{\lambda^*}(-1)$ is equal to :

Let $g:(0,\infty)\to\mathbb R$ be a differentiable function such that
$$\int\left(\frac{x(\cos x-\sin x)}{e^x+1}+\frac{g(x)\left(e^x+1-xe^x\right)}{\left(e^x+1\right)^2}\right)dx=\frac{x\,g(x)}{e^x+1}+c,$$
for all $x>0$, where $c$ is an arbitrary constant. Then :

Let $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be two functions defined by $f(x)=\log_e\left(x^2+1\right)-e^{-x}+1$ and $g(x)=\dfrac{1-2e^{2x}}{e^x}$. Then, for which of the following range of $\alpha$, the inequality
$$f\left(g\left(\frac{(\alpha-1)^2}3\right)\right)>f\left(g\left(\alpha-\frac53\right)\right)$$
holds ?

Let $y=y(x)$ be the solution of the differential equation $(x+1)y'-y=e^{3x}(x+1)^2$, with $y(0)=\dfrac13$. Then, the point $x=-\dfrac43$ for the curve $y=y(x)$ is :

Water is being filled at the rate of 1 cm$^3$/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 cm$^2$/sec) at which the wet conical surface area of the vessel increases is

If the angle made by the tangent at the point $(x_0,y_0)$ on the curve $x=12(t+\sin t\cos t)$, $y=12(1+\sin t)^2$, $0<t<\dfrac\pi2$, with the positive $x$-axis is $\dfrac\pi3$, then $y_0$ is equal to:

If the line $y=4+kx$, $k>0$, is the tangent to the parabola $y=x-x^2$ at the point $P$ and $V$ is the vertex of the parabola, then the slope of the line through $P$ and $V$ is:

Let $f(x)=\left|(x-1)\left(x^2-2x-3\right)\right|+x-3$, $x\in\mathbb R$. 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 ______.

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

Let S be the set of all the natural numbers, for which the line $\dfrac xa+\dfrac yb=2$ is a tangent to the curve $\left(\dfrac xa\right)^n+\left(\dfrac yb\right)^n=2$ at the point $(a,b)$, $ab\ne0$. Then :

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

Cosider 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 :