First Derivative Test

Use the sign chart for f'(x).  There is(are) ...

• a local maximum at x = -2.
• a local minimum at
  x = -2 and a local maximum at x = 4.
• a local maximum at
  x = -2 and a local minimum at x = 4.
• no extrema.

Over what intervals is f(x) decreasing?

• (-∞, -1) ∪ (1, ∞)

• (-∞, -√3) ∪ (0, √3)

• (-1, 1)

• (-√3, 0) ∪ (√3, ∞)

True or false: all critical points are extrema

• True

• False

If  $f'\left(a\right)=0$  and  $f'\left(x\right)$  changes from positive to negative at  $x=a$  , then  $f\left(x\right)$  has 

• A relative maximum at x=a

• A relative minimum at x=a

• No relative extrema at x=a

• A vertical tangent line at x=a

If  $f'\left(a\right)=0$  and  $f'\left(x\right)$  changes from negative to positive at  $x=a$  , then  $f\left(x\right)$  has

• a relative maximum at x=a

• a relative minimum at x=a

• no relative extrema at x=a

• a vertical tangent line at x=a

The function  $f\left(x\right)=-x^3+3x^2-5$  has a relative maximum value of _____________ at x=_____________.

• 0; -5

• -5; 0

• 2; -1

• -1; 2

Identify the critical points of the following function:
g(x)=2x³-3x²

• x=-1,1
• x=0,0
• x=0,-1
• x=0,1

Find the open interval(s) where the function is decreasing.

 $y=-\frac{x^2}{2x+2}$  

•  $\left(-2,-1\right),\left(-1,0\right)$  

•  $\left(-\infty,-8\right),\left(4,\infty\right)$  

•  $\left(-\infty,-2\right),\left(0,\infty\right)$  

•  $\left(-\frac{2}{3},-\frac{1}{3}\right),\left(-\frac{1}{3},\frac{1}{3}\right)$  

For the function given, identify the point(s) of relative maxima.

 $y=-\left(x+6\right)^{\frac{2}{3}}$  

• x = -24

• x = -6

• x = -2

• No relative maxima

For the function given, identify the point(s) or relative minima.

 $y=\cos\left(x\right)\ ;\ \left[0,2\pi\right]$  

•  $x=\pi$  

•  $x=0$  

•  $x=\left\{-\frac{\pi}{4},\frac{3\pi}{4},-\pi,\pi\right\}$  

• No points of relative minima

Tags

CCSS.HSF.TF.A.4