Posted by cher on Sunday, February 26, 2012 at 7:10pm.
Consider a differentiable functionf having domain all positive real numbers, and for which it is known that f'(x)=(4x)^3 for x.0
a. find the xcoordinate of the critical point f. DEtermine whether the point is a relative maximum or minomum, or neither for the fxn f. justify your answer.
b) find all intervals on which the graph of f is concave down. jusity answer.
c. given that f(1)=2 determine the fxn f.
i got a but how do you do b and c?

calculus  Reiny, Sunday, February 26, 2012 at 7:27pm
c)
if f'(x) = (4x)^3
then f(x) = (1/2)(4x)^2 + c
given f(1) = 2
2 = (1/2)(41)^2 + c
2 = (1/2)(1/9) + c
c = 2  1/18 = 35/18
f(x) = (1/2)(4x)^2 + 35/18
f''(x) = 3(4x)^4 (1) = 3/(4x)^4
the graph is concave up when f''(x) > 0
the graph is concave down when f''(x) < 0
since (4x)^4 is always positive, for x≠4
then 3/(4x)^4 is always positive.
So the curve is concave up for all x's in the domain.
here is what Wolfram thinks of your function
http://www.wolframalpha.com/input/?i=%281%2F2%29%284x%29%5E2+%2B+35%2F18
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