Given that $f(x)$ is continuously differentiable on $a \le x \le b$ where $a < b, f(a) < 0$ and $f(b) > 0$,which of the following are always true?
$(i)$ $f(x)$ is bounded on $a \le x \le b$.
$(ii)$ The equation $f(x) = 0$ has at least one solution in $a < x < b$.
$(iii)$ The maximum and minimum values of $f(x)$ on $a \le x \le b$ occur at points where $f'(c) = 0$.
$(iv)$ There is at least one point $c$ with $a < c < b$ where $f'(c) > 0$.
$(v)$ There is at least one point $d$ with $a < d < b$ where $f'(d) < 0$.

Rolle's theorem holds for a monic quadratic polynomial $f(x)$ on the interval $[\alpha, \alpha + 3]$ where $f(\alpha) = 0$. Similarly, $g(x) = f(x) + 2$ also follows Rolle's theorem on the interval $[\beta, 3]$ where $g(3) = 0$, such that the value of $c$ (where $f'(c) = g'(c) = 0$) is the same for both $f(x)$ and $g(x)$. Then the value of $(f \circ g)(\alpha)$ is...

If $f(x)$ satisfies the conditions of Rolle's theorem in $[1, 2]$ and $f(x)$ is continuous in $[1, 2]$,then $\int_1^2 f'(x) dx$ is equal to

The value of $c$ in the Lagrange's mean value theorem for $f(x)=\sqrt{x-2}$ in the interval $[2,6]$ is

The position of a moving car at time $t$ is given by $f(t) = at^{2} + bt + c, t > 0,$ where $a, b,$ and $c$ are real numbers greater than $1.$ Then the average speed of the car over the time interval $[t_{1}, t_{2}]$ is attained at the point

Let $\psi_1:[0, \infty) \rightarrow \mathbb{R}$,$\psi_2:[0, \infty) \rightarrow \mathbb{R}$,$f:[0, \infty) \rightarrow \mathbb{R}$,and $g:[0, \infty) \rightarrow \mathbb{R}$ be functions such that $f(0)=g(0)=0$,$\psi_1(x)=e^{-x}+x$ for $x \geq 0$,$\psi_2(x)=x^2-2x-2e^{-x}+2$ for $x \geq 0$,$f(x)=\int_{-x}^{x}(|t|-t^2)e^{-t^2} dt$ for $x>0$,and $g(x)=\int_0^{x^2} \sqrt{t} e^{-t} dt$ for $x>0$.
$(1)$ Which of the following statements is $TRUE$?
$(A)$ $f(\sqrt{\ln 3})+g(\sqrt{\ln 3})=\frac{1}{3}$
$(B)$ For every $x>1$,there exists an $\alpha \in(1, x)$ such that $\psi_1(x)=1+\alpha x$
$(C)$ For every $x>0$,there exists a $\beta \in(0, x)$ such that $\psi_2(x)=2x(\psi_1(\beta)-1)$
$(D)$ $f$ is an increasing function on the interval $[0, \frac{3}{2}]$
$(2)$ Which of the following statements is $TRUE$?
$(A)$ $\psi_1(x) \leq 1$,for all $x>0$
$(B)$ $\psi_2(x) \leq 0$,for all $x>0$
$(C)$ $f(x) \geq 1-e^{-x^2}-\frac{2}{3}x^3+\frac{2}{5}x^5$,for all $x \in(0, \frac{1}{2})$
$(D)$ $g(x) \leq \frac{2}{3}x^3-\frac{2}{5}x^5+\frac{1}{7}x^7$,for all $x \in(0, \frac{1}{2})$