What is a stationary point and how do I find where they occur and distinguish between them?

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A stationary point is simply a point on a graph where the derivative=0. Ie, the rate of change of the curve at this point is 0 and therefore it is neither increasing or decreasing at this point

There are three types you need to know about:

1) A maximum: Here the derivative =0 and the second derivative <0.

2) A minimum: Here the derivative =0 and the second derivative >0

3) A point of inflection: Here the derivative and the second derivative =0

Note, the second derivative means the derivative of the first derivative!

General solution:

Suppose y=f(x)

and dy/dx=f'(x)

If at a point, say c, f'(c)=0 then there is a stationary point at this value of x.

Differentiate f'(x) to get the second derivative.

Plug in the value of c again and if the solution is..

0 - Point of inflection

Positive - Minimum turning point

Negative - Maximum turning point


y = x3 - 6x2 + 9x - 4

Find any stationary points and determine their nature.


dy/dx = 3x2- 12x + 9

At a stationary point, dy/dx=0

So 3x2- 12x + 9 = 0

3(x2- 4x + 3) = 0  

(x - 3)(x - 1) = 0

So stationary point at x = 3 and x = 1.

Now, to determine the nature of these..

f''(x) = 6x - 12

f''(3) = 18 - 12 = 6 therefore minimum turning point at x = 3

f''(1) = 6 - 12 = -6 therefore maximum turning point at x = 1

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