For the solution to part b) of this question, they've written that 0≤c≤1. From my understanding of Lagrange's Remainder Theorem, the interval for the c term arises from the Mean Value Theorem, thus shouldn't open intervals be used? 0<c<1. Have they written a mistake here?
Furthermore, if we were to consider their final step of working (which I am curious to know why it is even necessary—aren't they making interval estimate less accurate?): ½(e - e^-1) < ½(3-⅓) = 4/3, wouldn't this imply E < 4/3 × 1/(15•7!) and not E ≤ 4/3 × 1/(15•7!) ?