Second Derivative Test for Local Minimum

Q. If $f\left(x\right)=\left|\begin{array}{ccc}\cos 2x& \cos 2x& \sin 2x\\ -\cos x& \cos x& -\sin x\\ \sin x& \sin x& \cos x\end{array}\right|,$ then

1. $f^{\prime }\left(x\right)=0$ at exactly three points in $(-\pi ,\pi )$
2. $f^{\prime }\left(x\right)=0$ at more than three points in $(-\pi ,\pi )$
3. $f\left(x\right)$ attains its maximum at $x=0$
4. $f\left(x\right)$ attains its minimum at $x=0$

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 minimum 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. 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. 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. If $f"\left(x\right)>0\forall xϵR$ then for any two real numbers $x_{1}and{x}_2,(x_1\ne x_2)$

1. $f\left(\frac{x_1+x_2}{2}\right)>\frac{f\left(x_1\right)+f\left(x_2\right)}{2}$
2. $f\left(\frac{x_1+x_2}{2}\right)<\frac{f\left(x_1\right)+f\left(x_2\right)}{2}$
3. $f^{\prime }\left(\frac{x_1+x_2}{2}\right)>\frac{f^{\prime }\left(x_1\right)+f^{\prime }\left(x_2\right)}{2}$
4. $f^{\prime }\left(\frac{x_1+x_2}{2}\right)<\frac{f^{\prime }\left(x_1\right)+f^{\prime }\left(x_2\right)}{2}$

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 $f$.