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Maximum & Minimum of a function

Calculus HOME | Achievement Objectives | Gradient | Gradient Functions | Differentiation | Gradient at a point | Find point with gradient | Equation of Tangent | Second Derivitive | Coordinates of Max & Min | Increasing, Decreasing Functions | Applications | Kinematics with differentiation | Antidifferentiation | Finding 'C' | Kinematics with anti-differentiation | Rates of Change | Mixed Problems | Revision

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.

 

Class Notes | Blank Notes

Graphs: Ex15.04,

Maximum & Minimum: Ex15.05, Ex15.06

NuLake IAS p89, 90

Guru maths site

 

Maximum

At the maximum the gradient = 0
As we move across the maximum the gradient DECREASES
Positive gradient changing to negative
 
 
Mouse over the x values above
 
At the maximum the gradient = 0
and the derivative = 0)

 

To find the maximum,
find where the derivative = 0 (find x)

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Minimum

At the minimum the gradient = 0
As we move across the minimum the gradient INCREASES
Negative gradient changing to positive
 
 
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)

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Point of inflection

At a point of inflection the gradient = 0
As we move across the  gradient STAYS POSITIVE
Gradient decreases to 0 then increases again
 
 
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)

 

Example:

Find the maximum value of the curve y = -x2 - 6x + 8

Steps:

y = -x2 - 6x + 8

1) Differentiate to find gradient function

y' = -2x - 6

2) The derivative = 0 at a min or max
So solve y' = 0 to find the x value for the maximum

0 = -2x - 6
6 = -2x
-3 = x

3) Substitute the x value back to find y

y = -x2 - 6x + 8

 

y = -(-3)2 - 6x-3 + 8
y = -9 + 18 + 8
y = 15

Maximum value = 15
Coordinates of the maximum (-3,15)

 

 

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