For each function, determine the coordinates of the local extrema. Classify each point as a local maximum or a local minimum.

f(x)=x^3 - 6x

For each function, determine the coordinates of the local extrema. Classify each point as a local maximum or a local minimum.

g(x)=-4x^4+2x^2

For each function, determine the coordinates of the local extrema. Classify each point as a local maximum or a local minimum.

f(x)=-x^3+3x-2

For each function, determine the coordinates of the local extrema. Classify each point as a local maximum or a local minimum.

h(x)= 2x^2 + 4x + 5

For each function, determine the coordinates of any points of inflection.

\displaystyle f(x)= 2x^3-4x^2

For each function, determine the coordinates of any points of inflection.

\displaystyle f(x)= x^5 + 20x^2 + 5

For each function, determine the coordinates of any points of inflection.

f(x)=x^5-30x^3

For each function, determine the coordinates of any points of inflection.

f(x)=3x^5-5x^4-40x^3+120x^2

Consider the function f(x)=x^4-8x^3.

a) Determine if the function is even, odd, or neither.

b) Determine the domain of the function.

c) Determine the intercepts

d) Find and classify the critical points. Identify the intervals of increase and decrease, and state the intervals of concavity.

Sketch each function.

f(x)=x^3+1

Sketch each function.

h(x)=x^5+20x^2+5

Sketch each function.

k(x)=\displaystyle{\frac{1}{2}}x^4-2x^3

Sketch each function.

b(x)=-(2x-1)(x^2-x-2)

Consider the function f(x)=2x^3-3x^2-72x+7

• What is the maximum number of local extrema this function can have? Explain.
• What is the maximum number of points of inflection this function can have? Explain.
• Find and classify the critical points. Identify the intervals of increase and decrease, and state the intervals of concavity.
• Sketch the function.

Do a full sketch by showing how you got

• Horizontal and vertical Asymptotes
• x-intercepts
• Any critical points (max/min)
• Intervals of increase and decrease
• Points of inflection and concavity

\displaystyle h(x) = 3x^2 -27

Do a full sketch by showing how you got

• Horizontal and vertical Asymptotes
• x-intercepts
• Any critical points (max/min)
• Intervals of increase and decrease
• Points of inflection and concavity

\displaystyle f(x) = x^5 -2x^4 + 3

Do a full sketch by showing how you got

• Horizontal and vertical Asymptotes
• x-intercepts
• Any critical points (max/min)
• Intervals of increase and decrease
• Points of inflection and concavity

f(x)= 3x^3 +7x^2 + 3x -1

Do a full sketch by showing how you got

• Horizontal and vertical Asymptotes
• x-intercepts
• Any critical points (max/min)
• Intervals of increase and decrease
• Points of inflection and concavity

f(x)= 2x^3 -12x^2 + 18x -4

Do a full sketch by showing how you got

• Horizontal and vertical Asymptotes
• x-intercepts
• Any critical points (max/min)
• Intervals of increase and decrease
• Points of inflection and concavity

f(x)= 2x^4 -26x^2 +72

Do a full sketch by showing how you got

• Horizontal and vertical Asymptotes
• x-intercepts
• Any critical points (max/min)
• Intervals of increase and decrease
• Points of inflection and concavity

h(x)= 5x^3 -3x^5

Write an equation for a function that has a local minimum when x = 2 and a point of inflection at x = 0.

Explain how these conditions helped you determine the function.

Joshua was sketching the graph of the functions f(x) and g(x). He lost part of his notes, including the equations for the functions. Given the partial information below, sketch each function.

The domain of f is \mathbb{R}.

\displaystyle \lim_{x\to \infty} f(x) = -\infty and \displaystyle \lim_{x\to -\infty} f(x) = -\infty

The y-intercept is 2. The x-intercepts are -1, 3, and 5. There is a local extrema at x = 3.

Joshua was sketching the graph of the functions f(x) and g(x). He lost part of his notes, including the equations for the functions. Given the partial information below, sketch each function.

The polynomial function g has domain \mathbb{R}.

\displaystyle \lim_{x\to -\infty} g(x) = \infty

g is an odd function.

There are five x-intercepts, two of which are 4 and -22. The function passes through ( 1, 5), which is close to a local extremum.

For each function,

i) How does the function behave as x\to \infty? Explain your reasoning.

ii) Does the function have any symmetry? Explain.

iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.

iv) Find the point(s) of inflection and make a table showing the intervals of concavity.

\displaystyle g(x) = x^3 -27x

For each function,

i) How does the function behave as x\to \infty? Explain your reasoning.

ii) Does the function have any symmetry? Explain.

iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.

iv) Find the point(s) of inflection and make a table showing the intervals of concavity.

\displaystyle y = x^4 - 8x^2 + 16

For each function,

i) How does the function behave as x\to \infty? Explain your reasoning.

ii) Does the function have any symmetry? Explain.

iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.

iv) Find the point(s) of inflection and make a table showing the intervals of concavity.

\displaystyle k(x) = \frac{1}{1 -x^2}

For each function,

i) How does the function behave as x\to \infty? Explain your reasoning.

ii) Does the function have any symmetry? Explain.

iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.

iv) Find the point(s) of inflection and make a table showing the intervals of concavity.

\displaystyle f(x) = \frac{x}{x^2 + 1}

For each function,

i) How does the function behave as x\to \infty? Explain your reasoning.

ii) Does the function have any symmetry? Explain.

iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.

iv) Find the point(s) of inflection and make a table showing the intervals of concavity.

h(x) = \frac{x-4}{x^2}