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Max and Min

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Finding Maximum and Minimum values
by Differentiation


When the question asks to find the co-ordinates, you will be expected to state both  x and y values.
It does not matter whether it is a maximum or a minimum or just a point on the curve, you will still have to state both values.
You will find the co-ordinates by substituting the values back into the original equation, f(x).

A co-ordinate is written (x , y)


Increasing, Decreasing
or Stationary

f '(x) is negative   the function is decreasing
f '(x) is zero           the function is stationary (not changing)
f '(x) is positive     the function is increasing.

Maximum, Minimum
Points of Inflection

The value f '(x) is the gradient at any point but often we want to find the Turning or Stationary Point (Maximum and Minimum points) or Point of Inflection
These happen where the gradient is zero,  f '(x) = 0.

Critical Points include Turning points and Points where
' (x) does not exist.

f ''(x)  is negative   the function is maximum turning point
f ''(x) is zero            the function may be a point of inflection  
f ''(x) is positive      the function is
minimum turning point

Concave Up, Down
or Neither

f ''(x) is negative    the function is concave downwards
''(x) is zero            the function changing from concave
                                 downwards to upwards (or the other way around) 
''(x) is positive      the function is
concave upwards.

Click here for
Maximum and Minimum
Notes for Tests and Examinations

TablefilledClick here for instructions how to construct the table
to find Maximum, Minimum, Increasing and Decreasing intervals and Concavity

Here are eight steps to help you solve maximising and minimising
 word problems,
often called Optimisation Questions.

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