Question

Find a and b such that the function
f(x)=x2+4x+6 if x<-1
ax+b if x>-1

Answer 1

\[\begin{cases} & \text{ if } x<-1, x^2 + 4x+6 \\ & \text{ if } x>1, ax+b \end{cases}\]
It is probably meant that f(x) has to be both continuous and differentiable (smooth)
\[\text{ if } x=-1, x^2 + 4x+6=ax+b\]
and
\[\text{ if } x=-1, \frac{d(x^2 + 4x+6)}{dx} =\frac{d(ax+b)}{dx}\]
Now we evaluate this
so
\(1 - 4+6 = b-a\), so, \(b-a=3\)
and
\[\text{ if } x=-1, \frac{d(x^2 + 4x+6)}{dx} =\frac{d( ax+b)}{dx}\]
So
\(\text{ if } x=-1, 2x + 4 = a\), therefore, \( -2 + 4 = a\) or \(a = 2\)
this can be substituted into
\(b-a = 3\), so \(b -2= 2\), or \(b = 4\)

so we have
\(a = 2\), and \(b = 4\)

Answer 2

@oswaldo12
Sorry I made a typo
\(b-a = 3\), hence, \(b-2 = 3\), or \(b = 5\)
So we have
(a = 2\) and (b = 5\)