Below is the question that I have been working with:
And here is the solution to part c), the part that I am stuck on:
Here’s the thing, I understand why the first two factors are greater than zero (see the solution above), but, not why the third factor is negative when delta is added to x.
Looking at the second factor, if b plus e to the power of negative c times x plus minus delta in brackets is greater than zero then shouldn’t the latter term be greater than minus b when you manipulate the equality? Then doesn’t that mean the factor dealt with last in the solution (see picture) can never be negative? (It was stated earlier that b is greater than zero.)
The point of inflexion is at $P\left(-\frac{\log (b)}{c},\frac{a}{2 b}\right)$
$$f''(x)>0\to b e^{c x}-1>0$$
$$x>-\frac{\log (b)}{c}$$ Therefore at $P$ concavity changes and this proves that $P$ is a point of inflexion.