Graphical Interpretation of Relation between Continuity and Differentiability

Q.

If $2x,x+8,3x+1$ are in A.P. then the value of $x$ will be

1. $3$

2. $7$

3. $5$

4. $-2$

Q.

If $li{m}_{x\to -3}\frac{3x^2+ax+a-7}{x^2+2x-3}$, exists, then $a=?$

1. $10$

2. $15$

3. $-15$

4. $-10$

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 f(x) = $\{\begin{array}{cc}\frac{x-2}{3x^2-10x+8};& x\ne 2\\ \frac{1}{2};& x=2\end{array}$, then

यदि f(x) = $\{\begin{array}{cc}\frac{x-2}{3x^2-10x+8};& x\ne 2\\ \frac{1}{2};& x=2\end{array}$, तब

1. f ′(2) = $-\frac{3}{4}$
2. f ′(2) = $\frac{1}{2}$
3. f ′(2) = $-\frac{1}{4}$
4. f(x) is not differentiable at x = 2
   
   x = 2 पर f(x) अवकलनीय नहीं है