Graphical Interpretation of Relation between Continuity and Differentiability

Q. Let $f\left(x\right)=\{\begin{array}{cc}\text{max}\{|x|,x^2\},& |x|\le 2\\ 8-2|x|,& 2<|x|\le 4\end{array}$
Let $S$ be the set of points in the interval $(-4,4)$ at which $f$ is not differentiable. Then $S$ :

1. is an empty set
2. equals $\{-2,-1,1,2\}$
3. equals $\{-2,-1,0,1,2\}$
4. equals $\{-2,2\}$

Q. Let $f,g:[-1,2]\to R$ be continuous functions which are twice differentiable on the interval $(-1,2).$ Let the values of $f$ and $g$ at the points $-1,0$ and $2$ be as given in the following table:

	$x=-1$	$x=0$	$x=2$
$f\left(x\right)$	$3$	$6$	$0$
$g\left(x\right)$	$0$	$1$	$-1$

In each of the interval $(-1,0)$ and $(0,2)$ the function $(f-3g{)}^{\prime \prime }$ never vanishes. Then the correct statement(s) is(are)

1. $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly three solutions in$(-1,0)\cup (0,2)$
2. $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly one solution in $(-1,0)$
3. $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly one solution in $(0,2)$
4. $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly two solutions in $(-1,0)$ and exactly two solutions in $(0,2)$

Q. Let $f\left(x\right)=\{\begin{array}{cc}\text{max}\{|x|,x^2\},& |x|\le 2\\ 8-2|x|,& 2<|x|\le 4\end{array}$
Let $S$ be the set of points in the interval $(-4,4)$ at which $f$ is not differentiable. Then $S$ :

1. is an empty set
2. equals $\{-2,-1,1,2\}$
3. equals $\{-2,-1,0,1,2\}$
4. equals $\{-2,2\}$

Q. If $f\left(x\right)=\underset{0}{\overset{x}{\int }}{e}^{t^2}(t-2)(t-3)dt$ for all $x\in (0,\infty ),$ then

1. $f$ has a local maximum at $x=2$
2. $f$ is decreasing on $(2,3)$
3. There exists some $c\in (0,\infty )$ such that $f^{\prime \prime }\left(c\right)=0$
4. $f$ has a local minimum at $x=3$

Q. If $x=1$ is the point of extrema for $f\left(x\right)=\{\begin{array}{cc}x,& 0<x<1\\ g\left(x\right),& 1\le x<2\end{array},$ then which of the following(s) is(are) correct

1. $g\left(x\right)$ is decreasing at $x=1$ and $g\left(1\right)>1$
2. $g\left(x\right)$ is increasing at $x=1$ and $g\left(1\right)>1$
3. $g\left(x\right)$ is decreasing at $x=1$ and $g\left(1\right)<1$
4. $g\left(x\right)$ is increasing at $x=1$ and $g\left(1\right)<1$

Q. If $x=1$ is the point of extrema for $f\left(x\right)=\{\begin{array}{cc}x,& 0<x<1\\ g\left(x\right),& 1\le x<2\end{array},$ then which of the following(s) is(are) correct

1. $g\left(x\right)$ is increasing at $x=1$ and $g\left(1\right)<1$
2. $g\left(x\right)$ is decreasing at $x=1$ and $g\left(1\right)<1$
3. $g\left(x\right)$ is decreasing at $x=1$ and $g\left(1\right)>1$
4. $g\left(x\right)$ is increasing at $x=1$ and $g\left(1\right)>1$

Q. If $x=1$ is the point of extrema for $f\left(x\right)=\{\begin{array}{cc}x,& 0<x<1\\ g\left(x\right),& 1\le x<2\end{array},$ then which of the following(s) is(are) correct

1. $g\left(x\right)$ is increasing at $x=1$ and $g\left(1\right)>1$
2. $g\left(x\right)$ is decreasing at $x=1$ and $g\left(1\right)<1$
3. $g\left(x\right)$ is decreasing at $x=1$ and $g\left(1\right)>1$
4. $g\left(x\right)$ is increasing at $x=1$ and $g\left(1\right)<1$