Question 1195393
<br>
--------------------------------------------------------------------<br>
I need to rewrite my response -- in the question box the derivatives and second derivatives did not show up well, so originally I was working with all the wrong conditions....<br>
--------------------------------------------------------------------<br>
Note the problem does not ask you to find an equation -- it only asks you to sketch a graph.<br>
But even with that I am still stumped, like you, because I don't think the given conditions can be met.<br>
(1) Conditions b and c tell us that the graph has local minima at x=-4 and x=5, and a local maximum at x=0.  Note that means that condition f is superfluous.<br>
(2) Conditions d and e tell us the concavity changes from upwards to downwards at x=-1 and from downwards to upwards at x=2, so there are points of inflection at x=-1 and x=2.<br>
So conditions b, c, d, e, and f give us 3 local extrema and 2 points of inflection, suggesting a polynomial of degree 4.<br>
However, all of the given conditions can't be met with a polynomial of degree 4.<br>
Given local extrema ONLY at x=-4, x=0, and x=5, the equation of the derivative is<br>
f'(x) = {{{a(x+4)(x)(x-5) = ax^3-ax^2-20ax}}}<br>
Then the second derivative would be<br>
f''(x) = {{{3ax^2-2ax-20a}}}<br>
But that second derivative does not have zeros at x=-1 and x=2.<br>
So it appears the best we can do with this problem is to ignore some of the conditions, and -- without trying to find an equation -- sketch a graph that is...
concave up from -infinity to -1, with a local minimum at -4;
concave down from -1 to 2, with a local maximum at 0; and
concave up from  to infinity, with a local minimum at x=5.<br>
----------------------------------------------------------------<br>
Perhaps you will get other responses to your post from tutors who see the problem differently than I do....<br>