Unfortunately I have already thrown out the working that I did for your question....
oh well..
Part c. I think your answer is a bit wrong.
the points are (3, 39) and (3.001, 39.042) rounded more
the gradient between these points will be \(\frac{39.042-39}{3.001-3}\)
Can you see what you have done wrong?
It is an approximation because they actually asked you for the gradient of the tangent at (3, 39)
And you have found the gradient of the secant between (3,39) and (3.001, 39.042)
You could just as easily found the gradient of the secant between the points where x=3 and x=2.999. That would have been slightly different but also an acceptable estimation.
Part b
b. Given is that the differential quotient of f on the interval [0, a] equals 0. How do I calculate a?
This just means that the gradient of the line joining the 2 points, one where x=0 and the other point where x=a has a gradient of 0
\(f(0)=12\\ f(a)=2a^3-a^2-6a+12\\ so\\ \frac{f(a)-f(0)}{a-0}=0\\ \frac{2a^3-a^2-6a+12-12}{a-0}=0\\ \frac{2a^3-a^2-6a}{a}=0\\\)
Can you take it from there?