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Cherri (Parent) April 17 2016
John (Parent) March 21 2015
John (Student) March 20 2015
John (Student) March 21 2015
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!
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 = 1see more