Nayland College  Mathematics Home . Year 9 . Year 10 . Level 1 . Level 2 . L3 Statistics . L3 Maths . L3 Calculus . About . Links 
9  Maximum & minimum Applications
Using differentiation techniques to determine maximum values, optimal solutions, of minimum values. Max & Min applications Skills: Extract relevant information from a word problem, form an equation, differentiate and solve the problem.

Graphs: Ex15.04, Maximum & Minimum: Ex15.05, Ex15.06 NuLake IAS p89, 90 
At the maximum the gradient = 0As we move across the maximum the gradient DECREASES


Mouse over the x values above 
At the maximum the gradient = 0and the derivative = 0)
To find the maximum,find where the derivative = 0 (find x) 

At the minimum the gradient = 0As we move across the minimum the gradient INCREASES


Mouse over the x values above 
At the minimum the gradient = 0 (derivative = 0)To find the minimum, find where the derivative = 0 (find x) 

At a point of inflection the gradient = 0As we move across the gradient STAYS POSITIVE


Mouse over the x values above 
At the point of inflection the gradient = 0 (derivative = 0)To find this point, find where the derivative = 0 (find x) 

Find the maximum value of the curve y = x^{2}  6x + 8 

Steps: 
y = x^{2}  6x + 8 
1) Differentiate to find gradient function 
y' = 2x  6 
2) The derivative = 0 at a min or max 
0 = 2x  6 
3) Substitute the x value back to find y 
y = x^{2}  6x + 8 

y = (3)^{2}  6x3 + 8 
Maximum value = 15 
