The Mean Value Theorem

The Mean Value Theorem guarantees a point $c$ in $(-1, 1)$ with $f'(c) = \dfrac{f(1) - f(-1)}{2}$ for three of the functions below. For which one does it make no guarantee?

Find the value of $c$ in $(1, 4)$ whose existence the Mean Value Theorem guarantees for $f(x) = x^{2}$ on $[1, 4]$.

A car travels 150 km along a highway in exactly 1.5 hours. Its position is a differentiable function of time. According to the Mean Value Theorem, there must be some instant at which the car's speedometer reads exactly what value?

Find the point $c$ promised by the Mean Value Theorem for $f(x) = \ln x$ on the interval $[1, e]$.

Let $f(x) = x^{2} - 4\pi x + 3\sin x$. Show that there is a real number $c$ with $f'(c) = 0$.

How many real roots does the function below have?

A function $f$ satisfies $f'(x) = 3$ for every real $x$, and $f(1) = 2$. Find $f(5)$.

Use the Mean Value Theorem to show that $\sqrt{1 + x} < 1 + \dfrac{x}{2}$ for every $x > 0$.

How many real roots does the function below have?

For which values of the constant $k$ does $f(x) = x^{3} - 3x + k$ have exactly one real root? Enter the smallest positive integer $k$ with this property.