Question

I am having difficulty finding the correct solution to problem 1E-4 part a). The approach should be very similar to that of 1D-7, looking at the answer I do not understand the approach they use and why they use it.

Answer 1

We need to find values of a and b such that (1) the two expressions have the same value at x=0, and (2) the two expressions have the same derivative at x=0. The first part is so easy that it may be hard to recognize as a solution: when evaluated at x=0, everything drops out of both expressions except the constant 4. Therefore these expressions will always join up at point (0,4) no matter what values we use for a and b.

The second part isn't much harder. We need to make the derivatives of the two expressions match up, so we want
\[2ax+b=25x^{4}+12x ^{3}+14x+8\]
But remember, this is happening at x=0, so everything disappears except b=8. So the final answer is that the function is differentiable when b=8. It doesn't matter what value we choose for a because at x=0 the elements including a disappear in both f(x) and f'(x).