ASSIGNMENT 5 - SOLUTIONSProblem 1.Done in class.Problem 2.Takef(x) =x2, whose graph is a parabola. Drop it by 1 (so now the function isg(x) =x2-1). Slide it to the right by 1 (the function becomesh(x) = (x-1)2-1 =x2-2x.)This graph has all the required properties.Problem 3.By definition, concave upwards means increasing first derivative. Iffandgareincreasing, then so is their sumf+g= (f+g) . So the first statement is true. But the secondis false.For example, bothf(x) =x2andg(x) =x4+x2are concave upwards on [0,1] yetf(x)-g(x) =-x4is concave downwards on [0,1].Problem 4.To start off, the domain off(x) =x√9-xis (-∞,9].The first derivative isf(x) =x√9-x+x(√9-x) =√9-x+x121√9-x(9-x) =√9-x-x2√9-x=2(9-x)-x2√9-x=18-3x2√9-x=326-x√9-xThe critical numbers off(x) arex= 6 (f(6) = 0) andx= 9 (f(x) DNE). Both points are inthe domain off(x).i) Let’s use the 1st derivative test. Atx= 6,f(x) changes sign from + to-, sof(6) = 6√3is a relative maximum. Atx= 9 there is no change in the sign off(x) because we cannot gopast 9. So, strictly speaking, the 1st derivative test does not apply at this point. However, we
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