Find values for `a` and `b` that will make `f` continuous everywhere, if f(x) = { `-3x^2` if `x lt 1;` `ax + b` if `1lt=xlt=4;` `x + sqrt(x)` if `4...

Hello!

The formulas used to define `f(x)` are elementary functions and they are continuous on a given intervals. The only problem problematic points are the joint points `1` and `4.`

Even at these points `f` has limits from the left and from the right. And for `f` to be continuous at `1` and `4` it is necessary and sufficient for these limits to coincide. It is evident that

`lim_(x->1-) f(x) = -3,` `lim_(x->1+) f(x) = a+b,`

`lim_(x->4-) f(x) = 4a+b,` `lim_(x->4+) f(x) = 6.`

This gives the linear system for `a` and `b,` `a+b=-3` and `4a+b=6.` Solve it by substitution: `b=-3-a` and `4a-3-a=6,` thus `a=3` and `b=-6.` This is the (unique) answer.