Concavity

Q. The Intervals of concavity and convexity of $f\left(x\right)=\frac{\ln x}{x}$ are

1. Convex :$(e^{\frac{3}{2}},\infty )$
   
   
2. Convex : $(0,e^{\frac{3}{2}})$
3. Concave :$(e^{\frac{3}{2}},\infty )$
   
4. Concave : $(0,e^{\frac{3}{2}})$

Q.

$f\left(x\right)=-3x^2+2x+5$ is concave at

1. 3

2. 2

3. 1

4. -2

Q.

If a function f(x) is concave at x = a, then f”(a) < 0.

1. True

2. False

Q. The value(s) of $a,$ for which the curve $f\left(x\right)=x^4+3a{x}^3+6x^2+5$ is not situated below any of its tangent lines is/are

1. $-\frac{2}{3}$
2. $-\frac{1}{3}$
3. $\frac{3}{4}$
4. $\frac{4}{3}$

Q. Let $f\left(x\right)$ be an invertible function such that $f^{\prime }\left(x\right)>0$ and $f^{\prime \prime }\left(x\right)>0$ for all $x\in R,$ then which of the following is/are correct ?
(where $x_1,x_2,\cdots ,x_n$ are different points)

1. $\frac{\sum _{r=1}^{n}f\left(x_r\right)}{n}>f\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
2. $\frac{\sum _{r=1}^{n}f\left(x_r\right)}{n}<f\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
3. $\frac{\sum _{r=1}^n{f}^{-1}\left(x_r\right)}{n}<f^{-1}\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
4. $\frac{\sum _{r=1}^n{f}^{-1}\left(x_r\right)}{n}>f^{-1}\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$