Second Derivative Test for Local Minimum

Q. Let the function $f$ be defined by $f\left(x\right)=x\ln x$, for all $x>0$. Then

1. $f$ is increasing on $(0,e^{-1})$
2. $f$ is decreasing on $(0,1)$
3. The graph of $f$ is concave down for all $x$
4. The graph of $f$ is concave up for all $x$

Q. Which of the following is (are) true about the function $f\left(x\right)=-\frac{3}{4}{x}^4-8x^3-\frac{45}{2}{x}^2+105$ ?

1. x = 0 is point of local maxima
2. x = 0 is point of local minima
3. x = –3 is point of local minima
4. x = –5 is point of local maxima.
5. x = –5 is point of local minima.

Q. Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
(i) x² + 2y² − 2x + 12y + 10 = 0
(ii) x² + 4y² − 4x + 24y + 31 = 0
(iii) 4x² + y² − 8x + 2y + 1 = 0
(iv) 3x² + 4y² − 12x − 8y + 4 = 0
(v) 4x² + 16y² − 24x − 32y − 12 = 0
(vi) x² + 4y² − 2x = 0

Q. Let $f\left(x\right)$ be a polynomial of degree $5$ such that $x=\pm 1$ are its critical points. If $\underset{x\to 0}{lim}(2+\frac{f\left(x\right)}{x^3})=4$, then which of the following is not true ?

1. $f\left(1\right)-4f(-1)=4$.
2. $x=1$ is a point of maxima and $x=-1$ is a point of minimum of $f$.
3. $f$ is an odd function.
4. $x=1$ is a point of minima and $x=-1$ is a point of maxima of $f$.

Q. Let $f\left(x\right)$ be a polynomial of degree $5$ such that $x=\pm 1$ are its critical points. If $\underset{x\to 0}{lim}(2+\frac{f\left(x\right)}{x^3})=4$, then which of the following is not true ?

1. $f$ is an odd function
2. $f\left(1\right)-4f(-1)=3$
3. $x=1$ is a point of maxima and $x=-1$ is a point of minimum of $f$
4. $x=1$ is a point of minima and $x=-1$ is a point of maxima of $f$