f (x) = a(x ? h) 2 + k. (a) What familiar type of function is f? What informatio
ID: 2870592 • Letter: F
Question
f (x) = a(x ? h)2 + k.
(a) What familiar type of function is f? What information do you know about f just by looking at its form? (Think about the roles of a, h, and k.)
(b) Next we use some calculus to develop familiar ideas from a different perspective. To start, treat a, h, and k as constants and compute f'(x).
(c) Find all critical values of f . (These will depend on at least one of a, h, and k.)
(d) Assume that a < 0. Construct a first derivative sign chart for f .
(e) Based on the information you’ve found above, classify the critical values of f as maxima or minima.
Explanation / Answer
here the given function is
f(x)=a(x-h)2+k
f(x)=ax2-2ahx+(ah2+k)
(a)
we can say that it is a quadratic equation. at first look
we can decide that it passes (0,ah2+k)
(b)
f'(x)=2ax-2ah
f'(x)=2a(x-h)
means it tangent passes through (h,0)
(c)
for finding critical value
f'(x)=0
2a(x-h)=0
x=h
now puuting value oh x=h in the original equation
f(x)=ax2-2ahx+(ah2+k)
f(h)=ah2-2ah2+ah2+k=k
therefore critical point is
(h,k)
(d)
(e)
f''(x)=2a
since a<0
there fore
f''(x)<0
therefore critical point is point of maxima
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