First Derivative Test for Local Maximum

Q. A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are

1. 24
2. 32
3. 45
4. 60

Q. Let f(x) = $\{\begin{array}{c}1+sinx,x<0\\ x^2-x+1,x\ge 0\end{array}$. Then

1. f has a local maximum at x = 0
2. f has a local minimum at x = 0
3. f is increasing every where
4. f is decreasing everywhere

Q. The absolute difference between the greatest and the least values of the function $f\left(x\right)=x(\ln x-2)$ on $[1,e^2]$ is

1. $2$
2. $e$
3. $e^2$
4. $1$

Q. A small island is situated $5$ km from the nearest point $A$ on the straight shoreline of a large lake. If a swimmer on this island can swin at a rate of $2$ kmph and can walk $3$ kmph on the land then, how far away should he land from point $A$ in order to arrive at his house $11$ km down the shore from $A$ in the least time?

1. $2$ km,
2. $4\sqrt{5}$ km,
3. $2\sqrt{5}$ km
4. $4$ km.

Q. Let $f:R\to R$ be given by
$f\left(x\right)=\{\begin{array}{c}{x}^5+5x^4+10x^3+10x^2+3x+1,x<0;\\ x^2-x+1,0\le x<1;\\ \frac{2}{3}{x}^3-4x^2+7x-\frac{8}{3},1\le x<3;\\ (x-2){\log }_e(x-2)-x+\frac{10}{3},x\ge 3.\end{array}$
Then which of the following option is/are correct?

1. $f$ is increasing on $(-\infty ,0)$
2. $f^{\prime }$ has a local maximum at $x=1$
3. $f$ is onto
4. $f^{\prime }$ is NOT differentiable at $x=1$

Q. Statement-1: $e^{\pi }>{\pi }^e$
Statement-2: ${\pi }^{e\pi }>e^{e\pi }$

1. Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1
2. Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation of Statement-1
3. Statement-1 is true, Statement-2 is false
4. Statement-1 is false, Statement-2 is true

Q. Statement-1: $e^{\pi }>{\pi }^e$
Statement-2: ${\pi }^{e\pi }>e^{e\pi }$

1. Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1
2. Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation of Statement-1
3. Statement-1 is true, Statement-2 is false
4. Statement-1 is false, Statement-2 is true

Q.

The function $f\left(x\right)=x^2(x-2{)}^2$

1. Decreases on (0, 1) ∪ (2, $\infty$)

2. Increase on ($-\infty$, 0) $\cup$ (1, 2)

3. Has a local maximum value 0

4. Has local maximum value 1