Applications of Derivatives

Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be two non-constant differentiable functions. If

$$f'(x) = \left(e^{(f(x) - g(x))}\right) g'(x) \quad \text{for all } x \in \mathbb{R},$$

and $f(1) = g(2) = 1$, then which of the following statement(s) is (are) TRUE?

Let $f: (0, \pi) \to \mathbb{R}$ be a twice differentiable function such that

$$\lim_{t \to x} \frac{f(x)\sin t - f(t)\sin x}{t - x} = \sin^2 x \quad \text{for all } x \in (0, \pi).$$

If $f\left(\frac{\pi}{6}\right) = -\frac{\pi}{12}$, then which of the following statement(s) is (are) TRUE?

Which of the following options is the only CORRECT combination?

Which of the following options is the only CORRECT combination?

Which of the following options is the only INCORRECT combination?

If $f: \mathbb{R} \to \mathbb{R}$ is a twice differentiable function such that $f''(x) > 0$ for all $x \in \mathbb{R}$, and $f\left(\frac{1}{2}\right) = \frac{1}{2}$, $f(1) = 1$, then

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

If $f(x) = \begin{vmatrix} \cos(2x) & \cos(2x) & \sin(2x) \\ -\cos x & \cos x & -\sin x \\ \sin x & \sin x & \cos x \end{vmatrix}$, then

The normal to the curve $y(x-2)(x-3)=x+6$ at the point where the curve intersects the $y$-axis passes through the point :

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :

The tangent at the point (2, −2) to the curve, $x^2y^2-2x=4(1-y)$ does not pass through the point :

The function $f$ defined by $f(x)=x^3-3x^2+5x+7$, is :

A tangent to the curve, $y=f(x)$ at P(x, y) meets $x$-axis at A and $y$-axis at B. If AP : BP = 1 : 3 and $f(1)=1$, then the curve also passes through the point :

The least value of $\alpha\in\mathbb R$ for which $4\alpha x^2+\dfrac1x\ge1$, for all $x>0$, is

The total number of distinct $x\in[0,1]$ for which $\displaystyle\int_0^x\frac{t^2}{1+t^4}\,dt=2x-1$ is

Let $f(x)=\displaystyle\lim_{n\to\infty}\left(\frac{n^n(x+n)\left(x+\frac n2\right)\cdots\left(x+\frac nn\right)}{n!\,(x^2+n^2)\left(x^2+\frac{n^2}{4}\right)\cdots\left(x^2+\frac{n^2}{n^2}\right)}\right)^{x/n}$, for all $x>0$. Then

Let $f:\mathbb R\to(0,\infty)$ and $g:\mathbb R\to\mathbb R$ be twice differentiable functions such that $f''$ and $g''$ are continuous functions on $\mathbb R$. Suppose $f'(2)=g(2)=0$, $f''(2)\ne0$ and $g'(2)\ne0$. If $\displaystyle\lim_{x\to2}\frac{f(x)g(x)}{f'(x)g'(x)}=1$, then

Consider $f(x)=\tan^{-1}\left(\sqrt{\dfrac{1+\sin x}{1-\sin x}}\right)$, $x\in\left(0,\dfrac\pi2\right)$. A normal to y = f(x) at $x=\dfrac\pi6$ also passes through the point :

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = $x$ units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then :

If the tangent at a point P, with parameter t, on the curve $x=4t^2+3$, $y=8t^3-1$, $t\in\mathbb R$, meets the curve again at a point Q, then the coordinates of Q are :