How to find the maximum point of a graph

• May 2nd 2010, 01:33 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)

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, 01:50 PM
rubic
Equation:
f(x) = x(8-2x)(6x-2x)??
• May 2nd 2010, 01:54 PM
florx
Oops typo.

Corrected:
f(x) = x(8-2x)(6-2x)
• May 2nd 2010, 02:42 PM
rubic
Equation:
f(x) = x(8-2x)(6-2x) = 4x(4-x)(3-x) = \$\displaystyle 4x^{3}-28x^{2}+48x\$

at the turning points

\$\displaystyle d/dx = 0\$
\$\displaystyle d/dx = 12x^{2}-56x+48\$
roots are \$\displaystyle (7-sqr(13))/3\$ and \$\displaystyle (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