How to find the maximum point of a graph

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May 2nd 2010, 02:33 PM
florx

How to find the maximum point of a graph

Question: Find the maximum y value of the graph between 0 ≤ x ≤ 3

Equation:
f(x) = x(8-2x)(6-2x)

Answer:
24.26

How would you find the maximum y value of this graph? Is it not like a parabola so when I tried finding the midpoint and then using the quadratic formula it didn't give me the highest value. Please explain the process and show your work. Thanks.
May 2nd 2010, 02:50 PM
rubic

Equation:
f(x) = x(8-2x)(6x-2x)??
May 2nd 2010, 02:54 PM
florx

Oops typo.

Corrected:
f(x) = x(8-2x)(6-2x)
May 2nd 2010, 03:42 PM
rubic

Equation:
f(x) = x(8-2x)(6-2x) = 4x(4-x)(3-x) = $4x^{3}-28x^{2}+48x$

at the turning points
$d/dx = 0$
$d/dx = 12x^{2}-56x+48$
roots are $(7-sqr(13))/3$ and $(7+sqr(13))/3$
until now all we know is these are turning points,
The value f ”(x) will tell us whether the point is a maximum or a minimum or a point of inflection
differentiate again
f''(x)=24x-56= 8(3x-7)

f''((7+sqr(13))/3) = positive, hence it is min
f''((7-sqr(13))/3) = negative, so this is max
f((7-sqr(13))/3) = 24.26

Answer:
24.26
May 2nd 2010, 04:49 PM
hmmmm

you also need to check the global extrema, where the graph gets cut off e.g. at where x=3 and x=0 however in this case they are less than this value