Mock Test 1 — Differentiability

Let $f(x) = |\log_e x|$. At $x = 1$, the function is:

If $f(x) = \tan^{-1} x$, the value of $f''(0)$ is:

The derivative of $\sec^{-1} x$ with respect to $x$ for $|x| > 1$ is:

If $y = \ln(\sec x + \tan x)$, then $\frac{dy}{dx}$ is:

The function $f(x) = x^{1/3}$ has which point profile at $x = 0$?

If $x^2 + 2xy + y^3 = 4$, then $\frac{dy}{dx}$ at the point $(1, 1)$ equals:

The value of the derivative of $e^{x^3}$ with respect to $\ln x$ at $x = 1$ is:

If Rolle's Theorem holds for $f(x) = x^3 - ax^2 + bx$ on $[1, 2]$ with $c = \frac{4}{3}$, then the values of the constants are:

If $y = \sin(m \sin^{-1} x)$, then $(1-x^2)y_2 - xy_1$ equals:

Which of the following functions does NOT satisfy Rolle's Theorem on $[-1, 1]$?

Find the number of points of non-differentiability for $f(x) = |x| + |x-1| + |x-2|$ across the real line.

If $y = \tan^{-1}\left(\frac{\cos x + \sin x}{\cos x - \sin x}\right)$ where $x \in (0, \pi/4)$, find the value of $2\frac{dy}{dx}$.

Find the value of $c$ that satisfies LMVT for $f(x) = \ln x$ on the interval $[1, e]$.

Assertion (A): The function $f(x) = x |x|$ is differentiable at $x = 0$.
Reason (R): If a function is continuous at a point, it must be differentiable at that point.

Assertion (A): If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, and $f(a) = f(b) = 0$, then the derivative vanishes at least once inside the interval.
Reason (R): Rolle's Theorem guarantees a flat tangent position where the slope is zero.