List of top Mathematics Questions on Differentiability

Let $f(x)=|\log_e x|-|x-1|+5$. Statement 1: $f(x)$ is differentiable for all $x\in(0,\infty)$
Statement 2: $f(x)$ is increasing in $(1,\infty)$
Statement 3: $f(x)$ is decreasing in $(0,1)$
Which of the following is correct?

Find the derivative of the function $ f(x) = 3x^2 - 5x + 7 $.

If $ f(x) = |x| + |x - 1| $, then which of the following is correct?

Let the function $f(x) = (x^2 - 1)|x^2 - ax + 2| + \cos|x| $ be not differentiable at the two points $ x = \alpha = 2 $ and $ x = \beta $. Then the distance of the point $(\alpha, \beta)$ from the line $12x + 5y + 10 = 0$ is equal to:

Let the function, $f(x)$ = $\begin{cases} -3ax^2 - 2, & x < 1 \\a^2 + bx, & x \geq 1 \end{cases}$ Be differentiable for all $ x \in \mathbb{R} $, where $ a > 1 $, $ b \in \mathbb{R} $. If the area of the region enclosed by $ y = f(x) $ and the line $ y = -20 $ is $ \alpha + \beta\sqrt{3} $, where $ \alpha, \beta \in \mathbb{Z} $, then the value of $ \alpha + \beta $ is:

The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is.

Check the differentiability of the function $f(x) = |x|$ at $x = 0$.

The Boolean expression $ (\sim(p \land q)) \lor q $ is equivalent to:

$ \int_{\pi/11}^{9\pi/22} \frac{dx}{1 + \sqrt{\tan x}} $ is equal to:

The function f(x) = |x| + |1 − x| is:

If $ y = e^{{2}\log_e t} $ and $ x = \log_3(e^{t^2}) $, then $ \frac{dy}{dx} $ is equal to:

Let $ g : \mathbb{R} \rightarrow \mathbb{R} $ be a non-constant twice differentiable function such that $ g'\left(\frac{1}{2}\right) = g'\left(\frac{3}{2}\right) $. If a real-valued function $ f $ is defined as \[ f(x) = \frac{1}{2} \left[ g(x) + g(2 - x) \right], \] then

$\frac{dy}{dx}$ = $\frac{(1-x-y^2)}{y}$ and $x(1)=1$ , then $5x(2)$ is equal to____.

If \[y = \frac{\left(\sqrt{x} + 1\right)\left(x^2 - \sqrt{x}\right)}{x\sqrt{x} + x + \sqrt{x}} + \frac{1}{15}(3\cos^2 x - 5)\cos^3 x,\]then $96y'\left(\frac{\pi}{6}\right)$ is equal to:

Consider the function $ f : (0, \infty) \rightarrow \mathbb{R} $ defined by $f(x) = e^{-| \log_e x |}.$If $ m $ and $ n $ be respectively the number of points at which $ f $ is not continuous and $ f $ is not differentiable, then $ m + n $ is

Let $ g(x) $ be a linear function and \[ f(x) = \begin{cases} g(x), & x \leq 0 \\ \left( \frac{1 + x}{2 + x} \right)^{\frac{1}{x}}, & x > 0 \end{cases} \] is continuous at $ x = 0 $. If $ f'(1) = f(-1) $, then the value of $ g(3) $ is

Let $ a $ and $ b $ be real constants such that the function $ f $ defined by \[f(x) = \begin{cases} x^2 + 3x + a, & x \leq 1 \\ bx + 2, & x>1 \end{cases}\]be differentiable on $ \mathbb{R} $. Then, the value of $ \int_{-2}^{2} f(x) \, dx $ equals

Let $ f(x) = (x + 3)^2 (x - 2)^3 $, $ x \in [-4, 4] $. If $ M $ and $ m $ are the maximum and minimum values of $ f $, respectively in $[-4, 4]$, then the value of $ M - m $ is:

Check the differentiability of $ f(x) = \begin{cases} x^2 + 1, & 0 \leq x < 1 \\ 3 - x, & 1 \leq x \leq 2 \end{cases} $ at $ x = 1 $.

A bacteria sample of a certain number of bacteria is observed to grow exponentially in a given amount of time. Using the exponential growth model, the rate of growth of this sample of bacteria is calculated. The differential equation representing the growth is:

\[ \frac{dP}{dt} = kP, \] where $ P $ is the bacterial population.

Based on this, answer the following:

Evaluate: \[ \int_{0}^{\pi/2} \sin 2x \cos 3x \, dx \]

Check the differentiability of $ f(x) $ at $ x = 1 $, where: \[ f(x) = \begin{cases} x^2 + 1, & 0 \leq x < 1, \\ 3 - x, & 1 \leq x \leq 2. \end{cases} \]

The number of points, where the function $f: \mathbb{R} \to \mathbb{R}$, $$ f(x) = |x - 1|\cos|x - 2|\sin|x - 1| + (x - 3)|x^2 - 5x + 4|, $$ is NOT differentiable, is

For $x > 1$, if $(2x)^{2y} = 4e^{2x - 2y}$, then $\left( 1 +\log_{e} 2x\right)^{2} \frac{dy}{dx} $ is equal to :

If $x = 3 \,tan \,t $ and $y = 3 \,sec \,t$, then the value of $\frac{d^2 y}{dx^2}$ at $t = \frac{\pi}{4}$, is :