A child wants to build a tunnel using three equal boards, each 4 feet wide one f

Question

A child wants to build a tunnel using three equal boards, each 4 feet wide one for the top and one for each side as shown. The two sides are to be tilted at equal angles theta to the floor. What is the maximum cross-sectional area A that can be achieved? Suppose that y = f(x) is a continuous function defined on the interval from x = 0 to x = E. Below is a graph of f'(x), the derivative of f(x), which is defined at all points of [0, E] except at x = C. (a) Where is f(x) is increasing? Where is f(x) is decreasing? Where does f(x) have local extreme values? (b) Where is f(x) concave up? Where is f(x) concave down? Where does f(x) have inflection points? (c) Draw a possible graph of f(x) which uses all information given and deduced about f(x).

Explanation / Answer

a)
The f' graph given to us....
Clearly when f' is positive, f is increasing
When f' is negative, f is decreasing

So, inc : [0 , B) U (B , C) U (D , E]

decreasing : (C , D)

f has extreme values at x = C and D

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b)
Concavity :
We are given f' graph
Clearly when f' is decreasing, thats when f'' < 0 and conc down
and when f' is increasing, thats when f'' > 0 and conc up

So, conc up : [0 , A) U (B , C) u (C , E)

conc down : (A , B)

Inf pts : x = A , x = B

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