Question

Find a and b such that f is differentiable everywhere.

Answer 1

The function is given as

`\begin{gathered} f(x)=\lbrace8\cos x\text{ if x}<0\rbrace \\ \lbrace ax+b\text{ if x}\ge0\rbrace \end{gathered}`

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.

`8\cos (0)=8`
`a(0)+b=b`

For function to be continuous, b=8.

Now for the two functions first derivative has the same value wen x=0.

`-8\sin x=-8\sin 0=0`
`(ax+b)^(\prime)=a`

Now for these have the same value.

`a=0`

Hence the value of b is 8 and a is 0.