Math Methods - 2nd Derivative Tests and Sketching

Using the 1st and 2nd derivative, sketch the function -x³+2x²

Answer explanation

y' = -3x^2 + 4x

0 = -3x^2 + 4x

0 = (x)(-3x+4)

therefore flat at x=0, x=4/3

y'' = -6x + 4

y''(0) = 4, positive = min

therefore x=4/3 is max

y(0) = 0, y(4/3) = -3(4/4)^2+4(4/3)= 1.185

turning points at (0,0), (4/3, 1.185)

(0,0) is one x-int and y int

other x int (sub y=0 into y):

0 = -x^3+2x^2

0=(x)(x)(-x + 2)

therefore x

The concavity of a function is described by its _______________.

• first derivative
• second derivative
• third derivative
• expression

If (a,b) is a local maximum, then what will be true about f''(a)?

• It's positive
• It's negative
• It's zero
• Cannot be determined

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

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

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

• increasing
• decreasing
• concave up
• concave down

Tags

CCSS.HSF-BF.B.4A

For a function f(x), f'(-3) = 5 indicates f(x) is ___________ at x=-3.

• increasing
• decreasing
• concave up
• concave down

Which graph is the derivative of the top graph?

• A
• B
• C
• D

Tags

CCSS.HSF.IF.B.4

List all of the critical values (x-locations of the critical points) of the function whose derivative is graphed.

• {-3, -1, 1}
• {-3, -1}
• {-4, -2, 0}
• Cannot be determined