For each of the following functions:
Consider the function f \left( x \right) = 3 \left(x - 4\right)^{3} + 1.
By inspecting the equation and/or the graph of the function, state what type of stationary point the function has.
Use the first derivative to find the x-coordinate of the stationary point.
Find the second derivative at this point.
Hence confirm what type of stationary point the function has.
For each of the following functions:
Find the x-coordinates of the stationary points.
Determine the nature of the stationary points.
For each of the following functions:
Find the derivative.
Find the stationary point(s).
Classify each stationary point.
Consider the function f \left( x \right) = 2 x^{3} - 18 x^{2} + 54 x - 49.
Find the x-coordinate(s) of the stationary point(s).
Find the x-coordinate(s) of the point(s) of inflection.
Determine the nature of the stationary point(s).
Consider the function f \left( x \right) = 4 x^{3} + 8 x^{2} + 5 x + 2.
Find the x-coordinate(s) of the stationary point(s).
Find the x-coordinate(s) of the point(s) of inflection.
Consider the function y = 5 - \left(x - 4\right)^{4}.
Find the coordinates of the stationary point.
Determine the nature of the stationary point.
Explain why x = 4 is a turning point and not a point of inflection.
Consider the function y = \left(x^{2} - 5\right)^{3}.
Find y''.
Find the points of inflection.
Which of these points are horizontal points of inflection?
Consider the function f \left( x \right) = - 4 \sin \left(x + \dfrac{\pi}{6}\right) on the interval 0 \leq x \leq 2 \pi.
Find the coordinates of any turning points.
Find an expression for f'' \left( x \right).
Hence classify the turning points.
For each of the following quadratic functions:
State whether the leading coefficient of f \left( x \right) is positive or negative.
Hence, determine the nature of the turning point.
Find f' \left( x \right).
Find f'' \left( x \right).
State whether the curve is concave up or down.
f \left( x \right) = x^{2} - 4 x + 9
f \left( x \right) = - x^{2} + 4 x - 9
Consider the function y = \left(x + 4\right)^{2} \left(x + 1\right) graphed below:
State the x-coordinates of the turning points of the function.
What is the gradient at these turning points?
State the x-coordinate of the point of inflection.
What sign is the gradient of the function at the point of inflection?
Sketch the graph of a possible gradient function.
Consider the function y = - \left(x + 13\right) \left(x - 2\right) \left(x - 11\right) graphed below.
State the x-coordinates of the turning points of the function.
State the x-coordinate of the point of inflection.
What sign is the sign of the gradient at the point of inflection?
Consider the function f \left( x \right) = \left( 3 x + 8\right) \left(x + 1\right).
Determine an equation for the gradient function f' \left( x \right).
State the coordinates of the turning point.
Determine the nature of the turning point.
Find the absolute minimum value of the function.
Consider the function f \left( x \right) = x^{3} - 12 x - 2.
Determine an equation for the gradient function f' \left( x \right).
State the coordinates of the turning points.
Determine an equation for the second derivative f'' \left( x \right).
Determine which turning point is a local minimum and which turning point is a local maximum.
Is - 18 the absolute minimum value of the function? Explain your answer.
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.
Determine which turning point is a local minimum and which turning point is a local maximum.
Is - 4 the absolute minimum value of the function? Explain your answer.
Consider the function f \left( x \right) = 10 x \sqrt{x + 3}.
Determine an equation for the gradient function f' \left( x \right).
Find the coordinates of the turning point.
Determine whether the point \left( - 2 , - 20 \right) is a minimum or maximum turning point.
Is - 20 the absolute minimum?
Consider the function f \left( x \right) = -3 x^{2} + 12 x + 2.
Find f' \left( x \right).
Find f'' \left( x \right).
Describe the rate of change of the gradient.
Are there any points of inflection on f(x)? Explain your answer.
Consider the function y = \left(x + 6\right)^{3}.
State the transformation that turns y = x^{3} into y = \left(x + 6\right)^{3}.
Find the point of inflection of y = x^{3}.
Find the point of inflection of y = \left(x + 6\right)^{3}.
Complete the following table of values:
| x | - 7 | - 6 | - 5 |
|---|---|---|---|
| y' | |||
| y'' | 0 |
Is the point of inflection, a horizontal point of inflection? Explain your answer.
For what values of x is the graph concave up?
Consider the function y = 4 x^{3} - 16 x^{2} + 4 x + 6.
Find y''.
Find the point of inflection.
Is the point of inflection, a horizontal or an ordinary point of inflection?
For what values of x is the graph concave down?
Consider the function y = x^{4} - 8 x^{3} - 9.
Find y''.
Find the points of inflection.
Complete the following table of values:
| x | - 2 | 0 | 2 | 4 | 6 |
|---|---|---|---|---|---|
| y' | |||||
| y'' | 0 | 0 |
Classify each point of inflection as an ordinary or horizontal point of inflection.
For what values of x is the graph concave up?
Consider the functiony = x e^{x}.
Find y'.
Find the x-coordinate of any stationary points.
Find the x-coordinate of any possible points of inflection.
Complete the table of value to confirm that there exists a point of inflection:
| x | - 3 | - 2 | - 1 |
|---|---|---|---|
| y'' |
For what values of x is the graph of y = x e^{x} concave up?
What type of stationary point is at x = -1?
Consider the function y = x^{5} - 3 x^{2}.
Find y''.
Find the x-coordinate of the potential point of inflection.
Is the point of inflection, a horizontal or an ordinary point of inflection?
For what values of x is the graph concave up?
For what values of x is the graph concave down?
Consider the function f \left( x \right) = \left(x - 8\right) \left(x - 5\right)^{2}.
Find f' \left( x \right).
Find the turning points.
Find f'' \left( x \right).
Classify the turning points.
For what values of x is the graph concave down?
Consider the function y = x \left(x - 3\right)^{2}.
Find the coordinates of the turning points.
Classify both stationary points.
Find the coordinates of the possible point of inflection.
Complete the table of values to prove that this is a point of inflection:
| x | 1 | 2 | 3 |
|---|---|---|---|
| y\rq\rq | 0 |
The first derivative of a certain function is f' \left( x \right) = 3 x^{2} + 9 x.
Determine the interval over which the function is increasing.
Determine the interval over which the function is decreasing.
Find f'' \left( x \right).
Determine the interval over which the function is concave up.
Determine the interval over which the function is concave down.
Find the x-coordinate of the maximum turning point.
Find the x-coordinate of the potential point of inflection.