Differentiability Questions (1063)

If the function \(g(x) = \begin{cases} k\sqrt{x+1}, & 0 \leq x \leq 3 \\ mx + 2, & 3

If the function \[f(x) = \begin{cases} -x, & x

If \(f(x) = |x|\) then \(f'(x) = \frac{|x|}{x}, x \neq 0\).State whether this statement is true or false.

The value of \(k\) for which the function \[f(x) = \begin{cases} \left(\dfrac{4}{5}\right)^{\frac{\tan 4x}{\tan 5x}}, & 0

Let \( f(x) \) be continuous and differentiable everywhere and defined as \[ f(x) = \begin{cases} ax^2 - b & x

If $\Delta_1 = \begin{vmatrix} x & b & b \\ a & x & b \\ a & a & x \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} x & b \\ a & x \end{vmatrix}$, then $\dfrac{d}{dx}(\Delta_1) = 3(\Delta_2)^{1/2}$... find the value (A: $3(\Delta_2)^{1/2}$):

Let $f(x) = \sin\!\left(\cos^{-1}(1-x^2)\right) + \cos\!\left(\sin^{-1}(2x-2x^3)\right)$. For $x\in(0,1/\sqrt{2})$, $f'(x)$ equals:

Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8$, $f'(2)=5$, $f'(x)\geq 1$ and $f''(x)\geq 4$ for all $x\in(1,6)$, then:

If f(x) = \log_2(\log x), then f'(x) at x = e is

Let f be a polynomial of degree 2018 such that f(1) = 1, f(2) = 0, f(3) = −5, f(−4) = 2. If f(x) is an even function then find the minimum number of points where f″(x) = 0.

\(\lim_{x \to 0} \frac{\sin\left(\pi \cos^2(\tan(\sin x))\right)}{x^2} = \)

Consider the function \( f(x) = |x^2 - 7x + 12|(x^2 - 7x + 10)(x^2 - 4x + 3) \). Then Rolle's theorem for \( f(x) \) is not applicable to which of the following range?

$\sin ax + \cos ax$ and $|\cos x| + |\sin x|$ are periodic functions of same fundamental period, if '$a$' equals