# Calculus : Differentiability and Maximizing Area

24) Let f be a differentibale function defined on the closed interval [a,b] and let c be a point in the open interval (a,b) such that.

I f'(c)=0

II f'(x)>0 when a<or equal to x<c, and

III f'(x)<0 when c<b<or equal to b.

Which is true? Then tell why others false.

a. F'(c)=0

b. F"(c)=0

c. F(c) is an abs. max. value of f on [a,b].

d. F(c) is an abs. min. value of f on [a,b].

e. F(x) has a point of inflection at x=c.

23) A rectangle with one side on the x axis and one side on the line x=2 has its upper left vertex on the graph of y=x^2. For what value of x does the area of the rectangle attain its maximum vlaue? Explain.

(Figure is an x/y axis. The curve is y=x^2 and where the function goes approx. straight up almost verticle draw a line to the x axis. Then make a rectangle inside the curve and line to the x axis.)

Let f(x) = x^3 + x. If h is the inverse function of f then h'(20) equals? Answer is 1/4 but give step by step solution to get answer please.

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#### Solution Preview

24. Here I think you mean F is f. From the condition, we know f'(c)=0 and f(c) is increasing from a to c and decreasing from c to b. Thus f(c) is the abs. max value of f on [a,b].

Thus, a,c is true, b,d,e is not true. From those which are not true, you only need to ...

#### Solution Summary

Differentiability of a function is investigated and maximum area of a rectangle is found.