Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(a) $3 / 4$.
(B) $4 / 3$.
(C) $3 / 2$.
(D) $2 / 3$.

10. Given the function:
$$f ( x ) = \left| 4 - x ^ { 2 } \right|$$
verify that it does not satisfy all the hypotheses of Rolle's theorem in the interval $[-3; 3]$ and that nevertheless there exists at least one point in the interval $[-3; 3]$ where the first derivative of $f(x)$ vanishes. Does this example contradict Rolle's theorem? Justify your answer thoroughly.
\footnotetext{Maximum duration of the examination: 6 hours. The use of scientific and/or graphing calculators is permitted provided they are not equipped with symbolic computation capability (O.M. no. 257 Art. 18 paragraph 8). The use of a bilingual dictionary (Italian – language of the country of origin) is permitted for candidates whose native language is not Italian. It is not permitted to leave the Institute before 3 hours have elapsed from the dictation of the theme. }

5. With a fence 2 metres long, one wants to enclose a surface having the shape of a rectangle topped by a semicircle, as in the figure: [Figure]
Determine the dimensions of the sides of the rectangle that allow enclosing the surface of maximum area.

Among all rectangular parallelepipeds with square base and volume $V$, determine whether the one with minimum total area also has minimum diagonal length.

Given the function $f_{a}(x) = x^{5} - 5ax + a$, defined on the set of real numbers, determine for which values of the parameter $a > 0$ the function has three distinct real zeros.

5. Determine the polynomial function of fourth degree $y = p ( x )$ knowing that, in a Cartesian coordinate system, its graph satisfies the following conditions: －it is tangent to the $x$-axis at the origin; －it passes through the point $( 1,0 )$; －it has a stationary point at $( 2 , - 2 )$.

13. If $f ( x ) = ( x 2 - 1 ) / ( x 2 + 1 )$, for every real number $x$, then the minimum value of $f$ :
(A) does not exist because $f$ is unbounded.
(B) is not attained even though $f$ is bounded
(C) is equal to 1
(D) is equal to - 1

29. The function $f ( x ) = \int - 1 x t$ (et-1)(t-1)(t-2)3(t-3)5dt has a local minimum at $x =$
(A) 0
(B) 1
(C) 2
(D) 3

22. Consider the following statements in S and R : $S$ : Both $\sin x$ and $\cos x$ are decreasing functions in the interval $( \sqcap / 2 , \sqcap )$ R : If a differentiable function decreases in an interval ( $\mathrm { a } , \mathrm { b }$ ), then its derivative also decreases in (a, b). Which of the following is true :
(A) Both S and R are wrong
(B) Both S and R are correct, but R is not the correct explanation of S .
(C) S is correct and R is correct explanation for S .
(D) S is correct and R is wrong.

28. Let $f ( x ) = \left\{ \begin{array} { c l l } | x | & \text { for } 0 < | x | \leq 2 \\ 1 & \text { for } & x = 0 . \end{array} \right.$
Then at $\mathrm { x } = 0 , \mathrm { f }$ has :
(A) A local maximum
(B) no local maximum
(C) a local minimum
(D) no extremum

34. $\sin - 1 ( x - x 2 / 2 + x 3 / 4 - \ldots ) + \cos - 1 ( x 2 - x 4 / 2 + x 6 / 4 - \ldots ) = n / 2$ for $0 < | x | < \sqrt { } ( 2$, ) then $x$ equals :
(A) $\frac { 1 } { 2 }$
(B) 1
(C) $- 1 / 2$
(D) - 1

9. The length of a longest interval in which the function $3 \sin x - 4 \sin ^ { 3 } x$ is increasing, is
(A) $\pi / 3$
(B) $\pi / 2$
(C) $3 \sqcap / 3$
(D) $\Pi$

21. If $f ( x ) = \int _ { x } { } ^ { 2 ( x 2 + 1 ) } e ^ { - t 2 } d t$, then $f ( x )$ increases in:
(a) $\quad ( 2,2 )$
(b) no value of $x$
(c) $\quad ( 0 , ¥ )$
(d) $( - ¥ , 0 )$

17. Prove that $\sin \mathrm { x } + 2 \mathrm { x } \geq \frac { 3 \mathrm { x } \cdot ( \mathrm { x } + 1 ) } { \pi } \forall \mathrm { x } \in \left[ 0 , \frac { \pi } { 2 } \right]$. (Justify the inequality, if any used).
Sol. Let $\mathrm { f } ( \mathrm { x } ) = 3 \mathrm { x } ^ { 2 } + ( 3 - 2 \pi ) \mathrm { x } - \pi \sin \mathrm { x }$ $\mathrm { f } ( 0 ) = 0 , \mathrm { f } \left( \frac { \pi } { 2 } \right) = - \mathrm { ve }$ $\mathrm { f } ^ { \prime } ( \mathrm { x } ) = 6 \mathrm { x } + 3 - 2 \pi - \pi \cos \mathrm { x }$ $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) = 6 + \pi \sin \mathrm { x } > 0$ $\Rightarrow \mathrm { f } ^ { \prime } ( \mathrm { x } )$ is increasing function in $\left[ 0 , \frac { \pi } { 2 } \right]$ ⇒ there is no local maxima of $\mathrm { f } ( \mathrm { x } )$ in $\left[ 0 , \frac { \pi } { 2 } \right]$ ⇒ graph of $\mathrm { f } ( \mathrm { x } )$ always lies below the x -axis [Figure] in $\left[ 0 , \frac { \pi } { 2 } \right]$. $\Rightarrow \mathrm { f } ( \mathrm { x } ) \leq 0$ in $\mathrm { x } \in \left[ 0 , \frac { \pi } { 2 } \right]$. $3 \mathrm { x } ^ { 2 } + 3 \mathrm { x } \leq 2 \pi \mathrm { x } + \pi \sin \mathrm { x } \Rightarrow \sin \mathrm { x } + 2 \mathrm { x } \geq \frac { 3 \mathrm { x } ( \mathrm { x } + 1 ) } { \pi }$.

17. Prove that $\sin \mathrm { x } + 2 \mathrm { x } \geq \frac { 3 \mathrm { x } \cdot ( \mathrm { x } + 1 ) } { \pi } \forall \mathrm { x } \in \left[ 0 , \frac { \pi } { 2 } \right]$. (Justify the inequality, if any used).
Sol. Let $\mathrm { f } ( \mathrm { x } ) = 3 \mathrm { x } ^ { 2 } + ( 3 - 2 \pi ) \mathrm { x } - \pi \sin \mathrm { x }$ $\mathrm { f } ( 0 ) = 0 , \mathrm { f } \left( \frac { \pi } { 2 } \right) = - \mathrm { ve }$ $\mathrm { f } ^ { \prime } ( \mathrm { x } ) = 6 \mathrm { x } + 3 - 2 \pi - \pi \cos \mathrm { x }$ $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) = 6 + \pi \sin \mathrm { x } > 0$ $\Rightarrow \mathrm { f } ^ { \prime } ( \mathrm { x } )$ is increasing function in $\left[ 0 , \frac { \pi } { 2 } \right]$ ⇒ there is no local maxima of $\mathrm { f } ( \mathrm { x } )$ in $\left[ 0 , \frac { \pi } { 2 } \right]$ ⇒ graph of $\mathrm { f } ( \mathrm { x } )$ always lies below the x -axis [Figure] in $\left[ 0 , \frac { \pi } { 2 } \right]$. $\Rightarrow \mathrm { f } ( \mathrm { x } ) \leq 0$ in $\mathrm { x } \in \left[ 0 , \frac { \pi } { 2 } \right]$. $3 \mathrm { x } ^ { 2 } + 3 \mathrm { x } \leq 2 \pi \mathrm { x } + \pi \sin \mathrm { x } \Rightarrow \sin \mathrm { x } + 2 \mathrm { x } \geq \frac { 3 \mathrm { x } ( \mathrm { x } + 1 ) } { \pi }$.

16. If $p ( x )$ be a polynomial of degree 3 satisfying $p ( - 1 ) = 10 , p ( 1 ) = - 6$ and $p ( x )$ has maximum at $x = - 1$ and $p ^ { \prime } ( x )$ has minima at $x = 1$. Find the distance between the local maximum and local minimum of the curve.

18. $f ( x )$ is cubic polynomial which has local maximum at $x = - 1$. If $f ( 2 ) = 18 , f ( 1 ) = - 1$ and $f ^ { \prime } ( x )$ has local minima at $x = 0$, then
(A) the distance between $( - 1,2 )$ and $( \mathrm { a } , \mathrm { f } ( \mathrm { a } ) )$, where $\mathrm { x } = \mathrm { a }$ is the point of local minima is $2 \sqrt { 5 }$
(B) $\mathrm { f } ( \mathrm { x } )$ is increasing for $\mathrm { x } \in [ 1,2 \sqrt { 5 } ]$
(C) $\mathrm { f } ( \mathrm { x } )$ has local minima at $\mathrm { x } = 1$
(D) the value of $\mathrm { f } ( 0 ) = 5$
Sol. (B), (C) The required polynomial which satisfy the condition is $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 4 } \left( 19 \mathrm { x } ^ { 3 } - 57 \mathrm { x } + 34 \right)$ $\mathrm { f } ( \mathrm { x } )$ has local maximum at $\mathrm { x } = - 1$ and local minimum at $\mathrm { x } = 1$ [Figure]
Hence $\mathrm { f } ( \mathrm { x } )$ is increasing for $\mathrm { x } \in [ 1,2 \sqrt { 5 } ]$.

20. $\mathrm { f } ( \mathrm { x } ) = \left\{ \begin{array} { l l } \mathrm { e } ^ { \mathrm { x } } , & 0 \leq \mathrm { x } \leq 1 \\ 2 - \mathrm { e } ^ { \mathrm { x } - 1 } , & 1 < \mathrm { x } \leq 2 \\ \mathrm { x } - \mathrm { e } , & 2 < \mathrm { x } \leq 3 \end{array} \right.$ and $\mathrm { g } ( \mathrm { x } ) = \int _ { 0 } ^ { \mathrm { x } } \mathrm { f } ( \mathrm { t } ) \mathrm { dt } , \mathrm { x } \in [ 1,3 ]$ then $\mathrm { g } ( \mathrm { x } )$ has
(A) local maxima at $\mathrm { x } = 1 + \ln 2$ and local minima at $\mathrm { x } = \mathrm { e }$
(B) local maxima at $\mathrm { x } = 1$ and local minima at $\mathrm { x } = 2$
(C) no local maxima
(D) no local minima
Sol. (A), (B) $\mathrm { g } ^ { \prime } ( \mathrm { x } ) = \mathrm { f } ( \mathrm { x } ) = \begin{cases} \mathrm { e } ^ { \mathrm { x } } & 0 \leq \mathrm { x } \leq 1 \\ 2 - \mathrm { e } ^ { \mathrm { x } - 1 } & 1 < \mathrm { x } \leq 2 \\ \mathrm { x } - \mathrm { e } & 2 < \mathrm { x } \leq 3 \end{cases}$ $\mathrm { g } ^ { \prime } ( \mathrm { x } ) = 0$, when $\mathrm { x } = 1 + \ln 2$ and $\mathrm { x } = \mathrm { e }$ $g ^ { \prime \prime } ( x ) = \left\{ \begin{array} { c c } - e ^ { x - 1 } & 1 < x \leq 2 \\ 1 & 2 < x \leq 3 \end{array} \right\}$ $\mathrm { g } ^ { \prime \prime } ( 1 + \ln 2 ) = - \mathrm { e } ^ { \ln 2 } < 0$ hence at $\mathrm { x } = 1 + \ln 2 , \mathrm {~g} ( \mathrm { x } )$ has a local maximum $\mathrm { g } ^ { \prime \prime } ( \mathrm { e } ) = 1 > 0$ hence at $\mathrm { x } = \mathrm { e } , \mathrm { g } ( \mathrm { x } )$ has local minimum. ∵ $\mathrm { f } ( \mathrm { x } )$ is discontinuous at $\mathrm { x } = 1$, then we get local maxima at $\mathrm { x } = 1$ and local minima at $\mathrm { x } = 2$.

Section - C

Comprehension I

There are $n$ urns each containing $n + 1$ balls such that the $i$ th urn contains $i$ white balls and ( $n + 1 - i$ ) red balls. Let $u _ { i }$ be the event of selecting ith urn, $\mathrm { i } = 1,2,3 \ldots , \mathrm { n }$ and w denotes the event of getting a white ball.

Let $f(x) = 2 + \cos x$ for all real $x$. STATEMENT-1: For each real $t$, there exists a point $c$ in $[t, t+\pi]$ such that $f'(c) = 0$. because STATEMENT-2: $f(t) = f(t+2\pi)$ for each real $t$.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True

Let $f(x) = x^x$ for $x > 0$. Then $f$ is
(A) increasing on $(0, \infty)$
(B) decreasing on $(0, \infty)$
(C) increasing on $(0, 1/e)$ and decreasing on $(1/e, \infty)$
(D) decreasing on $(0, 1/e)$ and increasing on $(1/e, \infty)$

Let $f(x) = \frac{x}{\sqrt{a^2+x^2}} - \frac{d-x}{\sqrt{b^2+(d-x)^2}}$, where $a$, $b$, and $d$ are positive constants. Then
(A) $f$ is an increasing function of $x$
(B) $f$ is a decreasing function of $x$
(C) $f$ is neither increasing nor decreasing function of $x$
(D) $f'$ is not a monotonic function of $x$

Let $f(x) = 2x^3 - 3x^2 - 12x + 4$. Then
(A) $f$ has a local maximum at $x = -1$ and a local minimum at $x = 2$
(B) $f$ has a local minimum at $x = -1$ and a local maximum at $x = 2$
(C) $f$ has local minima at $x = -1$ and at $x = 2$
(D) $f$ has local maxima at $x = -1$ and at $x = 2$

The total number of local maxima and local minima of the function $$f ( x ) = \begin{cases} ( 2 + x ) ^ { 3 } , & - 3 < x \leq - 1 \\ x ^ { 2 / 3 } , & - 1 < x < 2 \end{cases}$$ is
(A) 0
(B) 1
(C) 2
(D) 3

Let $f ( x )$ be a non-constant twice differentiable function defined on $( - \infty , \infty )$ such that $f ( x ) = f ( 1 - x )$ and $f ^ { \prime } \left( \frac { 1 } { 4 } \right) = 0$. Then,
(A) $f ^ { \prime \prime } ( x )$ vanishes at least twice on $[ 0,1 ]$
(B) $f ^ { \prime } \left( \frac { 1 } { 2 } \right) = 0$
(C) $\quad \int _ { - 1 / 2 } ^ { 1 / 2 } f \left( x + \frac { 1 } { 2 } \right) \sin x d x = 0$
(D) $\int _ { 0 } ^ { 1 / 2 } f ( t ) e ^ { \sin \pi t } d t = \int _ { 1 / 2 } ^ { 1 } f ( 1 - t ) e ^ { \sin \pi t } d t$

Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Which of the following is true?
(A) $f ( x )$ is decreasing on $( - 1,1 )$ and has a local minimum at $x = 1$
(B) $f ( x )$ is increasing on $( - 1,1 )$ and has a local maximum at $x = 1$
(C) $f ( x )$ is increasing on $( - 1,1 )$ but has neither a local maximum nor a local minimum at $x = 1$
(D) $f ( x )$ is decreasing on $( - 1,1 )$ but has neither a local maximum nor a local minimum at $x = 1$