Second Derivative Test for Local Maximum

Q.

The point in the interval $[0,2\pi ]$ where $f\left(x\right)=e^{x}sinx$ has maximum slope, is

1. $\frac{\pi }{4}$
2. $\frac{\pi }{2}$

3. $\pi$

4. $3\frac{\pi }{2}$

Q. Let $a_1,a_2,a_3,...$ be an A.P. with $a_6=2.$Then the common difference of this A.P., which maximises the product $a_1{a}_4{a}_5,$ is :

1. $\frac{2}{3}$
2. $\frac{8}{5}$
3. $\frac{6}{5}$
4. $\frac{3}{2}$

Q. The maximum slope of the curve $y=\frac{1}{2}{x}^4-5x^3+18x^2-19x$ occurs at the point :

1. $(2,9)$
2. $(2,2)$
3. $(3,\frac{21}{2})$
4. $(0,0)$

Q.

If the local maximum of f(x) = sin2x - xϵ(0, π) is at x = a, then find the value of 36 $\frac{a}{\pi }$

___

Q. If one corner of a long rectangular sheet of paper of width $1$ unit is folded over, so as to reach the opposite edge of the sheet, then

1. minimum length of the crease is $\frac{3\sqrt{3}}{4}$
2. minimum length of the crease is $\frac{\sqrt{3}}{4}$
3. reduced width of the paper is $\frac{1}{2}$
4. reduced width of the paper is $\frac{1}{4}$

Q. The maximum area (in sq. units) of a rectangle having its base on the $x$-axis and its other two vertices on the parabola, $y=12-x^2$ such that the rectangle lies inside the parabola, is:

1. $36$
2. $32$
3. $20\sqrt{2}$
4. $18\sqrt{3}$

Q.

f(x) and f’(x) are differentiable at x = c. Which of the following is the condition for f(x) to have a local maximum at x = c, if f’(c) = 0

1. f”(c) > 0

2. f”(c) < 0

3. f”(c) = 0

4. None of the above.

Q. If one corner of a long rectangular sheet of paper of width $1$ unit is folded over, so as to reach the opposite edge of the sheet, then

1. minimum length of the crease is $\frac{3\sqrt{3}}{4}$
2. minimum length of the crease is $\frac{\sqrt{3}}{4}$
3. reduced width of the paper is $\frac{1}{2}$
4. reduced width of the paper is $\frac{1}{4}$