Differential Calculus Questions (112)

Let $f(x)$ is a polynomial function and $(f(x))^2 + (f'(x))^2 = 0$, then find $\lim_{x \to 0}\frac{f(x)}{f'(x)}\left[\frac{f'(x)}{f(x)}\right]$, (where [.] denotes greatest integer function) is_____.

A lane of width $27m$ runs at right angle out of a road of $64m$. The maximum length of a pole which can be carried from the road to the lane keeping it horizontal is $L$, then $\sqrt[3]{L}$ equals to _____.

Water is filled at rate $\pi$ cm$^3$/s in right circular conical vessel (vertex up) of height 5 cm and diameter 8 cm. When water height is 3 cm, rate of increase of wet conical surface area is (cm$^2$/s)

Let $f$ and $g$ be continuously differentiable functions such that $f(0) = 0, f'(0) = 2$ and $g(x) = f(-x + f(x))). The value of $g'(0)$ equals.

The function $f: (a, \infty) \to R$ where $R$ denotes the range corresponding to the given domain, with rule $f(x) = 2x^3 - 3x^2 + 6$ will have an inverse provided:

The least value of $'u'$ for which the equation, $\frac{4}{\sin x} + \frac{1}{1 - \sin x} = u$ has atleast one solution on the interval $(0, \pi/2)$ is:

If $f(x) = \frac{e^{x^2}(1+4x)^{1/2}}{\ln(1-x^2)}$ for $x \neq 0$, then $f$ has:

If $f(x) = 4x^3 - x^2 - 2x + 1$ and $g(x) = \begin{cases} \min\{f(t): 0 \leq t \leq x\}, & 0 \leq x \leq 1 \\ 3 - x & , 1 h''(x) h(x)$ for each $x \in J$. Then:

Given $f(x) = \frac{e^x - \cos 2x - x}{x^2}$ for $x \in \mathbb{R} - \{0\}$, $\{x\}$ is fractional part function $$g(x) = \begin{cases} f\{x\} & n 1)$ is equal to :

Water is filled at rate $\pi$ cm$^3$/s in right circular conical vessel (vertex up) of height 5 cm and diameter 8 cm. When water height is 3 cm, rate of increase of wet conical surface area is (cm$^2$/s)

Which of the following functions is differentiable at $x=0$?

Let $f(x)=\displaystyle\int e^x(x-1)(x-2)\,dx$. Then $f(x)$ decreases in the interval