Mean Value Theorem

If $f(x)$ is twice differentiable in $[0,2],f(0)=0,f(1)>0$ and $f(2)=0$, then which of the foll...

A function defined as f(x) = \frac{1 - \cos 4x}{x^2} for x \neq 0 and f(0) = m is continuous; find m...

Using L'Hopital's rule or expansion, the limit \lim_{x \to 0} \frac{e^x - 1 - x}{x^2} evaluates to w...

Evaluate \lim_{x \to 0} \frac{x - \sin x}{x - \tan x}, a ratio of two third-order vanishing trigonom...

Define f(x) = x^2 for x \le 1 and f(x) = ax + b for x > 1; find a, b making f differentiable everywh...

For the greatest integer function, at how many points in the open interval (0,3) is g(x) = [x] + [-x...

Consider h(x) = |x - 1| + |x + 1| on the real line; at how many points does h fail to be differentia...

Let q(x) = \max{x, x^3} for real x; determine the set of points where q is not differentiable.

For f(x) = x^3 - 3x on the closed interval [-\sqrt{3}, \sqrt{3}], how many values of c satisfy Rolle...

Let p(x) = x^2 \sin(1/x) for x \neq 0 and p(0) = 0; which statement about p at the origin is correct...

If y = x^x for x > 0, then the derivative dy/dx evaluated through logarithmic differentiation equals...

Applying the Lagrange mean value theorem to f(x) = \ln x on [1, e], the point c guaranteed by the th...