Concavity and The Second Derivative Test Lesson

​Concavity and the Second Derivative Test

​"Life is always in session. That is why you always have to pay attention to the lesson."
By. Mr. C.

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Match

Match the following. #8

• Finding a derivative means...

• Evaluating an integral means...

• The 1st Derivative Test uses...

• The 2nd Derivative Test uses...

• Critical points are...

finding the slope of a tangent line.

finding the area under a curve.

the first derivative to determine if a critical point is a local extrema and if so what type.

the 2nd derivative to determine if a critical point is a local extrema and if so what type.

where the 1st derivative equals 0 or is undefined.

• finding the slope of a tangent line.

• finding the area under a curve.

• the first derivative to determine if a critical point is a local extrema and if so what type.

• the 2nd derivative to determine if a critical point is a local extrema and if so what type.

• where the 1st derivative equals 0 or is undefined.

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Multiple Choice

The concavity of a function is described by its _______________. #1

• first derivative
• second derivative
• third derivative
• expression

Multiple Choice

What will be true at an inflection point?  (select the best answer) #2

• f(x)=0
• f'(x)=0
• f''(x)=0
• The function is undefined

Multiple Choice

For a function g(x), g''(3)=-8 indicates that g(x) is ____________ at x=3. #3

• increasing
• decreasing
• concave up
• concave down

Multiple Choice

If f''(x)>0, then what will be true about f'(x) over that same interval? #4

• f '(x) is constant

• f '(x) is increasing

• f '(x) is decreasing

• f '(x) must have an inflection point in that interval

Multiple Choice

f(x) is pictured. Inflection points are most likely at which x values? #5

• x=0 and 1.5

• x=2 and 2.5

• x=2.25

• x=1

Multiple Choice

Which of the following describes an interval of f(x) that is both decreasing and concave up? #6

• f '(x) < 0 and f "(x) < 0

• f '(x) < 0 and f "(x) > 0

• f '(x) > 0 and f "(x) > 0

• f '(x) > 0 and f "(x) < 0

Multiple Choice

Find the second derivative of the function: #7
f (x) =  2x - 5x⁶

• f ''(x)= 2 - 30x
• f ''(x) =  2-30x⁵
• f ''(x) = -30x⁵
• f ''(x) = -150x⁴

Multiple Choice

Where is the point of inflection for the function  $$f\left(x\right)=x^3+6x^2$$  ? #9

• $$x=0$$  

• $$x=-4$$  

• $$x=-2$$  

• $$x=2$$  

Multiple Choice

Determine the concavity of the function: $$f\left(x\right)=x^4-6x^3-60x^2+5x-12$$   Determine when the function is concave DOWN (interval notation). #10

• $$\left(-\infty,-2\right)\ and\ \left(5,\infty\right)$$  

• $$\left(-\infty,-2\right)$$  

• $$\left(-2,5\right)$$  

• $$\left(5,\infty\right)$$  

Multiple Choice

If the function f has a critical number at $$x=3$$ and $$f''\left(3\right)=5$$ which of the following would be true? #11

• f has a local maximum at x=3

• f has a local minimum at x=3

• f has neither a local maximum or minimum at x=3

• there is not enough information to know if there is a local maximum or minimum at x=3

Multiple Choice

What does the concept of concavity mean in calculus? #12

• Concavity in calculus refers to the color of the graph

• Concavity in calculus refers to the number of x-intercepts on the graph

• Concavity in calculus refers to the length of the graph

• Concavity in calculus refers to the shape of a graph, specifically whether the graph is curving upwards (concave up) or downwards (concave down).

Multiple Choice

How do you use the second derivative to find inflection points? #14

• By finding the integral of the function and using the concavity test to determine potential inflection points.

• By finding the second derivative of the function and using the concavity test to determine potential inflection points.

• By finding the first derivative of the function and using the concavity test to determine potential inflection points.

• By finding the average rate of change of the function and using the concavity test to determine potential inflection points.

Multiple Choice

What are inflection points in calculus? #13

• Inflection points are points where the curve intersects the x-axis

• Inflection points are points where the curve intersects the y-axis

• Inflection points are points where the curve is at its highest or lowest

• Inflection points in calculus are points on a curve where the concavity changes, indicating a change in the direction of the curve's curvature.

Multiple Choice

Which of the following uses the 2nd Derivative Test for relative extrema correctly? #20

• f'(c)=0.

  f''(c)>0.

  Thus, f(x) must have a relative maximum at x=c.

• f'(c)=0.

  f''(c)<0.

  Thus, f(x) must have a relative minimum at x=c.

• f'(c)=0.

  f''(c)>0.

  Thus, f(x) must have a relative minimum at x=c.

• f''(c)=0.

  f''(x) changes from + to - at x=c.

  Thus, f(x) has a point of inflection at x=c.

Multiple Choice

Find the point(s) of inflection  (if it exists) of the function:

$$f\left(x\right)=2x^3+6x^2-4$$  #26

• Where x = -1

• Where x = 1

• Where x =  - 2

• Where x = 0

• No point of inflection

Multiple Choice

For a function f(x), f''(4)=0 indicates that x=4 is _____________. #27

• an inflection point
• a critical point
• a relative maximum
• a relative minimum

Multiple Choice

Find the concavity of the function $$f\left(x\right)=17e^{\left(6x\right)}$$ when $$x=3$$ ...#33

• the function is concave upward

• the function is concave downward

Multiple Choice

Find the second derivative of f(x) = x² + e^x  - cosx. #39

• f"(x) = 2 + e^x + cosx
• f"(x) = 2x + e^x + cosx
• f"(x) = 2x + xe^x - cosx
• f"(x) = 2x + e^x + sinx

Multiple Select

Find the point(s) of inflection (if it exists) of the graphed function. #46

• No point of inflection

• Points C, D and E

• Points A, C, E and G

• Points A, B, C, D, E, F and G

Multiple Select

Find the point(s) of inflection (if it exists) of the graphed function. #47

• No point of inflection 

• Points A and B

• Points  A, B and Z

• Points X, A, Y, B, Z

Match

Match the following functions with their derivatives. #55

• f(x) = sin x

• f(x) = cos x

• f(x) = tan x

• f(x) = sec x

• f(x) = csc x

f'(x) = cos x

f'(x) = -sin x

f'(x) = sec ² x

f'(x) = sec x tan x

f'(x) = -csc x cot x

• f'(x) = cos x

• f'(x) = -sin x

• f'(x) = sec ² x

• f'(x) = sec x tan x

• f'(x) = -csc x cot x

Multiple Choice

$$\frac{d}{dx}f\left(g\left(x\right)\right)$$ #56

• $$f'\left(g\left(x\right)\right)g'\left(x\right)$$

• $$f'\left(x\right)g\left(x\right)+f\left(x\right)g'\left(x\right)$$

• $$f'\left(g'\left(x\right)\right)$$

• $$f'\left(g'\left(x\right)\right)g\left(x\right)$$

Match

Match the following derivative rule with the function. #52

• f(x) = $$x^2\sin x$$

• f(x) = $$\frac{x^2-x}{x+1}$$

• f(x) = $$\sqrt[]{x}$$

• f(x) = $$\sqrt[]{\sin\ \left(2x\right)}$$

• f(x) = 2x cos (3x-1)

product rule

quotient rule

power rule

chain rule

product & chain rules

• product rule

• quotient rule

• power rule

• chain rule

• product & chain rules