26.05 Stationary points (Extended)
State the type of point that matches the following descriptions:
A point where the curve changes from decreasing to increasing.
A point where the curve changes from increasing to decreasing.
A point where the tangent is horizontal and the concavity changes about the point.
For each of the following functions:
Find the derivative.
Find the coordinates of any stationary points.
Classify each stationary point.
f \left( x \right) = 134 - 300 x + 240 x^{2} - 64 x^{3}
f \left( x \right) = \left(x^{2} - 9\right)^{2} + 4
Consider the parabola with equation y = 5 + x - x^{2}.
Find the coordinates of the vertex of the parabola.
State the gradient of the tangent to the parabola at the vertex.
What type of stationary point is at the vertex of this parabola?
Consider the function f \left( x \right) = x^{2} + 4 x + 9.
Find an equation for the gradient function f' \left( x \right).
State the interval in which the function is increasing.
State the interval in which the function is decreasing.
Find the coordinates of the stationary point.
Classify the stationary point.
Consider the function f \left( x \right) = 4 x^{3} + 5 x^{2} - 4.
Find an equation for the gradient function f' \left( x \right).
Find the coordinates of the stationary points.
Complete the table of values:
| x | -2 | -\dfrac{5}{6} | -\dfrac{1}{2} | 0 | 1 |
|---|---|---|---|---|---|
| f'\left( x \right) | 0 | 0 |
Hence determine the:
Local minimum
Local maximum
Is - 4 the absolute minimum value of the function? Explain your answer.
Consider the function f \left( x \right) = 3 x^{2} - 54 x + 241.
Find f' \left( x \right).
Find the x-coordinate of the stationary point.
Classify the stationary point.