draw a possible graph of a continuous function y=f(x) that satisfies the three conditions:
i.) f'(x) > 0 for 1<x<3
ii.) f'(x) < 0 for x < 1 and x > 3
iii.) f'(x) = 0 for x = 1 and x = 3
Let's try polynomoials.
f'(x) has roots for x=1 and x=3, take -(x-1)*(x-3) for this. It is parabola branches down, i.e. it is positive between 1 and 3 (its roots) and negative outside. Great.
So f'(x) = -(x-1)*(x-3) = -x^2 + 4x - 3 and
f(x) = -(1/3)*x^3 + 2*x^2 - 3x + C (any C, I choose C=0).
The graph of f is here: https://www.desmos.com/calculator/qjx2sh9m3w
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