Hello, Mathhelpz!
These are not simple problems.
1) The number of inflection points for the graph of: .
in the interval
The answer is 8. We must solve: .
We have: .
Then: .
. .
And we must solve: .
I recommend a graphing calculator to find the intersections of:
. .
We have: .
It is a "negative" cubic with xintercepts 0 and 2.
. . The graph is tangent to the xaxis at
We have: .
It is an upopening parabola with xintercepts 0 and 2.
Those integrals are comparing the areas under the curves.
The question becomes: .When is the cubic above the parabola?
. . I tried to sketch the graphs, but failed abysmally.
The curves intersect at
The cubic is above the parabola on the interval
But it is also above the parabola for:
3) What does: . look like?
Or rather what does it do?
The expression: .
. . says: the tangent to the graph of at has slope 2.
The graph might look like this: Code:

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  +    +          
 2
