Answer: Provided the function, we have to answer the questions (a) to (f), the function is as follows:
[tex]f(x)=x^3+3x^2-105x+24\rightarrow(1)[/tex]
(a) The first derivative of (1) is as follows:
[tex]\begin{gathered} \text{ Using the power rule:} \\ \\ f^{\prime}(x)=3x^2+6x-105 \end{gathered}[/tex]
(b) The second derivative is calculated using the same method as in part (a), only the difference is that it is calculated twice:
[tex]f^{^{\prime}^{\prime}}(x)=6x+6[/tex]
(c) The interval where the function is increasing can be found by inspecting the graph of the original function or the equation (1):
The plot is as follows:
The interval on which the function is increasing is:
[tex]x=[-\infty,-7]\text{ and }x=[5,\infty][/tex]
(d) Similarly, the decreasing interval is as follows:
[tex]x=[-7,5][/tex]
(e) Concave upward:
[tex]x=[-7,+\infty][/tex]
(f) Concave downward:
Similarly, the concave downward is on the following interval:
[tex]x=[-\infty,-7][/tex]
