Application of Derivatives

Critical Numbers

100

Find the critical numbers of the function f(x)=8x³+81x²-42x-8

x=7, x=1/4

200

Find the critical numbers of the function f(x)=12x⁴+4x³-12x²-10

x=0, x=1/2, x=-1

300

Find the critical numbers of the function f(x)=6x⁵+33x⁴-30x³+100

x=0, x=3/5, x=-5

400

Find the critical numbers of the function 

f(x)= x³-3x²+2

x=0, x=2

500

Find the critical numbers of the function

f(x)=5x/(x²-4)

no critical values 

vertical asymptote: +/- 2

horizontal asymptote:0

Maximum/Minimum

100

Find the max and min of the function 

f(x)=x³+6x²-15x+17

Min: (-5, 117)

Max: (1, -3)

200

Find the local extrema of the function  

f(x)= x³-3x+4

Local Minimum: (-1, 6)

Local Maximum: (1, 2)

300

Find the max and min of the function f(x)=-2x⁵+5x⁴+80x³+1

Min: (-4, 8449)

Max: (6, 8209)

400

Find the max and min of the function f(x)= x²+4x-9

Max: None 

Min: (-4, 9)

500

Find the max and min of the function

f(x)= x²+18x+72

Max: None

Min: (-9, -9)

Increasing/Decreasing

100

Find the intervals in which the function f(x)=2x²-4x is increasing and decreasing

Decreasing: (- ∞,1)

Increasing: (1, ∞)

200

Find the intervals in which the function 

f(x)= 2x³+6x²+8 is increasing and decreasing

Increasing: (- ∞,-2)U(0, ∞)

Decreasing:(-2,0)

300

Find the intervals of increasing and decreasing values for the following function: 

f(x)= 3x⁵-5x³+3

Increasing: (-∞,-1) U (1,∞)

Decreasing: (-1,1)

400

Find the intervals of increase and decrease for the following function: 

f(x)= x⁴+4x³-8x²+19

Increasing: (-4, 0)U(1, ∞)

Decreasing: (-∞, -4)U(0, 1)

500

Find the intervals in which the function 

f(x)= 3x³+9x²-27x+6 is increasing and decreasing

Increasing: (-∞, -3)U(1, ∞)

Decreasing: (-3, 1)

Point of Inflection

100

Find the point of inflection of the function:

 f(x)=x³-3x²-12x+7

POI: (1,-7)

200

Find the point of inflection of the function:

f(x)= 2x³+6x²+8

POI: (-1, 12)

300

Find the point of inflection of the following function: 

f(x)= 4x³-8x²-12x+15

POI: (2/3, 4.63)

400

Find the point of inflection of f(x)=x⁴-4x³-8x²+16

POI: 15/16 or -1/2

500

Find the point of inflection of the function 

f(x)=x⁶/30 - x⁵/20 -x⁴+3x+20

POI: -3, 4

Challenge

100

Find the max/mins, increasing/decreasing intervals, critical values,  and points of inflection of the function f(x)=3x⁴-4x³+2

Increasing: (-∞, 0)U(1, ∞)

Decreasing: (0,1)

Critical Values: x=0, x=1

Max: (0,2)

Min: (1,1)

POI: 0, 2/3

200

Find the max/mins, increasing/decreasing intervals, critical values, and points of inflection of the function f(x)=x³+9x²+15x-11

Increasing: (- ∞,-5)U(-1, ∞)

Decreasing:(-5,-1)

Critical values: x=-1, x=-5

Max: (-5,14)

Min: (-1, -18)

POI: (-3, 88)

300

Find the intervals of increasing and decreasing, critical values, extrema, and any points of inflection of the following function: 

f(x)= x³-2x+3

Increasing: (-∞, -2/3) U (2/3,∞)

Decreasing: (-2/3,2/3)

Critical Values: ±sqrt 2/3

Maximum: (-2/3,4.037) 

Minimum: (2/3, 1.963) 

POI: NONE  

400

Find the intervals of increasing and decreasing, critical values, extrema, and any points of inflection of the following function: 

f(x)= x³+6x²+3

Increasing: (-∞, -4)U(0, ∞)

Decreasing: (-4, 0)

Critical Values: x=0, x=-4)

Maximum: (-4, 35)

Minimum: (0, 3)

POI: (2, 35)

500

Find the intervals of increasing and decreasing, critical values, extrema, and any points of inflection of the following function: 

f(x)= x³+9x²+24x

Increasing: (-∞, -4) U (-2, ∞)

Decreasing: (-4, -2)

Critical Values: -4, -2

Maximum: (-4, -16)

Minimum: (-2, -20)

POI: (-3, -18)