Consider the given graph of the function
f \left( x \right) = - \left(x - 4\right)^{3} + 7:
State the x-coordinate of the stationary point.
State the domain for which f \left( x \right) is decreasing.
For each of the following functions graphed below:
State the x-coordinate of the stationary point(s).
State the domain for which f \left( x \right) is increasing.
State the domain for which f \left( x \right) is decreasing.
Consider the given graph of the function f \left( x \right):
State the x-coordinate of the stationary point.
State the domain for which f \left( x \right) is increasing.
State the domain for which f \left( x \right) is decreasing.
State the domain for which f \left( x \right) is constant.
Find the stationary point(s) for the following functions:
y=x^2-2x+3
y=6+4x-x^2
y=2x^3+9x^2-24x+1
y=x^3+3x^2+3x-6
y=x^2+2x-3
y=-x^2+6x-2
y=x^4-2x^2
Show that the following curves have no stationary points:
y=\dfrac{x}{x-1}
y=x\sqrt{x-1}
y=\dfrac{x}{\sqrt{x+1}}
y=3x-\dfrac{4}{x+2}
Determine whether following statements describe a maximum or a minimum turning point:
A point where the curve changes from increasing to decreasing.
A point where the curve changes from decreasing to increasing.
For each of the following functions:
Find the derivative.
Find the stationary point(s).
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
For each of the following quadratic equations:
Find the stationary point(s).
Classify each stationary point.
y=x^2+4x-2
y=-2x^2+8x+1
y=x^3-x^2-8x+1
y=x^3+5x^2+8x-4
y=x^4-8x^2
y=x^4-4x^3
y=x^2(x-1)^4
y=\dfrac{2x}{x^2+1}
y=3x+\dfrac{12}{x+1}
The function y = a x^{2} - b x + c passes through the points (5, - 42) and (4, - 66) and has a maximum turning point at x = 3. Find the following:
\dfrac{dy}{dx}
a
c
b
Consider the cubic function y = x^{3} - a x^{2} + b x + 11, which has stationary points at x=2 and x=10. Find the following:
\dfrac{dy}{dx}
a
b
Consider the equation of the parabola y = 3 x^{2} - 18 x + 24.
Find the x-intercepts.
Find the y-intercept.
Find \dfrac{dy}{dx}.
Find the stationary point.
Classify the stationary point.
Sketch the graph of the parabola.
Sketch the linear function for which f \left( 0 \right) = 1 and f' \left( 2 \right) = 3.
Sketch the quadratic function, f \left( x \right), that satisfies the following conditions:
- f \left( 0 \right) = - 18
- f \left( 3 \right) = 0
- f \left( 6 \right) = 6
- f' \left( 6 \right) = 0
f' \left( x \right) > 0 for x < 6
- f \left( 0 \right) = 16
- f \left( 2 \right) = 0
- f \left( 5 \right) = - 9
- f' \left( 5 \right) = 0
f' \left( x \right) < 0 for x < 5
- f \left( 0 \right) = 5
- f \left( - 2 \right) = 0
- f' \left( 3 \right) = 0
f' \left( x \right) > 0 for x < 3
Sketch a cubic function, f \left( x \right), that satisfies the following conditions:
- f' \left( - 5 \right) = 0
f' \left( x \right) > 0 for all other values of x.
- f \left( 0 \right) = 7
- f \left( - 2 \right) = 0
- f \left( - 4 \right) = - 1
- f' \left( - 4 \right) = 0
f' \left( x \right) > 0 for x < - 4
f' \left( x \right) > 0 for x > - 4
- f' \left( 2 \right) = 0
- f' \left( - 3 \right) = 0
f' \left( x \right) < 0 for - 3 < x < 2
f' \left( x \right) > 0 elsewhere
Sketch a quartic function, f \left( x \right), that satisfies the following conditions:
Consider the given graph of the gradient function f' \left( x \right) = 2 x + 6:
State the x-intercept of the gradient function.
Is the gradient of f \left( x \right) for x > - 3 positive or negative?
Is the gradient of f \left( x \right) for x < - 3 positive or negative?
State the feature of f \left( x \right) that this x-intercept represent.
The function f \left( x \right) has a derivative given by f' \left( x \right) = 6 \left(x - 2\right) \left(x - 7\right). A graph of the derivative function is shown:
State the x-intercept(s) of the gradient function.
State the kind of feature at the point \left(2, 77\right) on the graph of f \left( x \right).
State the kind of feature at the point \left(7, - 48 \right) on the graph of f \left( x \right).
The function f \left( x \right) has a derivative given by f' \left( x \right) = 3 \left(x + 5\right)^{2}. A graph of the derivative function is shown:
State the x-intercept of the gradient function.
State the kind of feature at the point \left( - 5 , 2\right) on the graph of f \left( x \right).
Consider the gradient function f' \left( x \right) = 12 \left(x + 4\right)^{2} \left(x + 7\right).
Graph the gradient function.
State the kind of feature at the point \left( - 7 , - 1617 \right) on the graph of f \left( x \right).
State the kind of feature at the point \left( - 4 , - 1536 \right) on the graph of f \left( x \right).