The function is given as
[tex]\begin{gathered} f(x)=\lbrace8\cos x\text{ if x}<0\rbrace \\ \lbrace ax+b\text{ if x}\ge0\rbrace \end{gathered}[/tex]
We have to make sure that function is differentiable everywhere for that the function is continuous everywhere .
So for the piecewise continuous function given check the continuity at x=0.
[tex]8\cos (0)=8[/tex][tex]a(0)+b=b[/tex]
For function to be continuous, b=8.
Now for the two functions first derivative has the same value wen x=0.
[tex]-8\sin x=-8\sin 0=0[/tex][tex](ax+b)^{\prime}=a[/tex]
Now for these have the same value.
[tex]a=0[/tex]
Hence the value of b is 8 and a is 0.
