Let $f$ be a continuous function on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, where $a<b$. It is known that there exists $c$ in $(a,b)$ such that

$$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}.$$

Determine $c$ for $f(x)=2x^3-3x^2+7x-2$ on $[a,b]$ with $a=1$ and $b=6$.

(a) Consider the function $f(x)=\frac{\ln x}{x}$. Find the critical point(s) of $f$ and say whether $f$ has a maximum or minimum at that point.
(b) Prove that for integers $a>b\ge 3$, we must have $a^b<b^a$.

A new game show on TV has 100 boxes numbered $1,2,\dots ,100$, each containing a mystery prize. The prizes are of different types, $a,b,c,\dots$, in decreasing order of value, and are distributed randomly among the boxes. The most expensive item is of type $a$, a diamond ring, and there is exactly 1 of these. You are told that the number of items at least doubles as you move to the next category — there are at least twice as many items of type $b$ as of type $a$, at least twice as many items of type $c$ as of type $b$ and so on.
You ask for the type of item in box 45. Instead of being given a direct answer, you are told that there are 31 items of the same type as box 45 in boxes 1 to 44 and 43 items of the same type as box 45 in boxes 46 to 100.
Which of the following statements can be true?

The following graph shows the performance of students in an exam. The marks scored by every student are an multiple of five. The $j^{th}$-percentile $u^{\ast }$ for a discrete data $x_1,x_2,\dots ,x_n$ is defined as follows. Let $x_{(1)},x_{(2)},\dots ,x_{(n)}$ be the ordering of the data in ascending order. Let $t=\frac{jn}{100}$ and let $k$ be an integer such that $k\le t<(k+1)$ and let $s=t-k$. Then $u^{\ast }=x_{(k)}+s\ast (x_{(k+1)}-x_{(k)})$. Here, $x_{(n+1)}$ is defined to be $x_{(n)}$.

Based on the information presented in the graph, answer the following questions.
(a) Compute the 10th percentile of marks.
(b) Is the median score higher than the mean score?

Common Description: Description for the next two questions
The probability density function of a normal distribution with mean $\mu$ and variance ${\sigma }^2$ is of the form $$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-0.5(\frac{x-\mu }{\sigma }{)}^2},-\infty <x<\infty .$$ Let $X$ be a random variable with mean $E(X)=\mu$ and variance $Var(X)={\sigma }^2$. Let $X_1,X_2,\dots ,X_n$ be $n$ independently sampled values of $X$, and let $\overline{X}=\frac{1}{n}{\sum }_i{X}_i$ be the sample mean. The central limit theorem states that if the sample size $n$ is large enough, then $\overline{X}$ approximately follows the normal distribution with mean $\mu$ and variance ${\sigma }^2/n$. That is, $\overline{X}\approx N(\mu ,{\sigma }^2/n)$. This in turn implies that $$Y=\frac{\sqrt{n}(\overline{X}-\mu )}{\sigma }\approx N(0,1).$$ Let $Z$ be a random variable that follows the normal distribution with mean 0 and variance 1. For any real number $a$ let $\Phi (a)=P(Z<a)$ be the probability that $Z$ takes values smaller than $a$. Then $$\Phi (a)=P(Z<a)={\int }_{-\infty }^a\frac{e^{-u^2/2}}{\sqrt{2\pi }}du.$$ For solving the next two problems you may assume the following approximations: $\Phi (-2)=0.02,\Phi (-1)=0.16,\Phi (0)=0.5,\Phi (1)=0.84,\Phi (2)=0.98$.
The weekly number of sales $S$ at a certain car dealership is known to follow a probability distribution with mean $E(X)=10$ and variance $Var(X)=144$. A performance audit picks a random sample of 36 weekly sales figures $S_1,S_2,\dots ,S_{36}$ from the last two years. They find that the sample mean is 10 and the sample variance is 144. Use this information to answer the next two questions. You may assume that $n=36$ is a large enough sample size. (a) What is the probability that the average number of sales in a week will be more than 8?
(b) What is the total number of sales over the 36 weeks which were sampled?