**For continuity **

Critical points are x=1 and x=2 , so we will check there.

* 1)* $f(1^{+})$=$f(1)$=$f(1^{-})$

$1=a+b+c$ ----------$equation 1$

* 2)* $f(2^{+})$=$f(2)$=$f(2^{-})$

$4a+2b+c=2+d$ -------$equation 2$

**For derivability **

* 1)* $f'(1^{+})$=$f'(1^{-})$

$\frac{\mathrm{d} (x^{2})}{\mathrm{d} x}$=$\frac{\mathrm{d} (ax^{2}+bx+c)}{\mathrm{d} x}$

so $2x=2ax+b$ // put $x=1$

$2=2a+b$ ------$equation 3$

* 2) *$f'(2^{+})$=$f'(2^{-})$

$\frac{\mathrm{d} (ax^{2}+bx+c)}{\mathrm{d} x}$=$\frac{\mathrm{d} (x+d)}{\mathrm{d} x}$

so $2ax+b=1$ at $x=2$

$4a+b=1$ --------$equation 4$