Shapes of functions

Question 1

Where do the possible points of inflection occur?

Answer 1

When d^2/dx^2 = 0

Question 2

When is a curve concave?

Answer 2

If d^2y / dx^2 < 0
(same for max point)

Question 3

When is a curve convex?

Answer 3

If d^2y / dx^2 > 0
(same for min point)

Question 4

What do we always have to use to see if it is a point of inflection?

Answer 4

a change in sign test

Question 5

How would you find the points of inflection for y = x^3 ?

Answer 5

work out dy/dx which is 3x^2
work out d^2y / dx^2 which is 6x
Write statement “points of inflection occur when d^2y / dx^2 = 0
therefore 6x=0 –> x=0
work out d^2y/dx^2 when x= -0.1, 0 and 0.1
when x=-0.1, d^2y/dx^2=-0.6 (d^2y/dx^2 < 0 so concave)
when x=0, d^2y/dx^2=0
when x=0.1, d^2y/dx^2=0.6 (d^2y/dx^2 >0 so convex)
there is a change in sign therefore a point of inflection when x=0
dy/dx = 3x^2, when x=0, dy/dx=3(0) = 0 therefore it is a stationary point of inflection

Question 6

How would you find the points of inflection for y = x^3 -3x ?

Answer 6

dy/dx = 3x^2 - 3 –> d^2y/dx^2=6x
points of inflection occur when d^2y/dx^2 =0
6x=0 –> x=0
when x=-0.1, d^2y/dx^2=-0.6(<0 so concave)
when x=0. d^2y/dx^2=0
when x=0.1, d^2y/dx^2=0.6( >0 so convex)
change in sign, therefore point of inflection when x=0
dy/dx = 3x^2 - 3, when x=0 dy/dx=-3 therefore non stationary point of inflection

Question 7

Sketch the graph of y=x^3 - 3x, showing all turning points and point of inflection

Answer 7

dy/dx=3x^2 - 3 ,stationary points occur when dy/dx=0 –> 3x^2 - 3=0 –> x=±1
when x=-1, y=2
when x=1, y=-2
d^2y/dx^2=6x
when x=1, d^2y/dx^2=6, >0 so min point (1,-2)
when x=-1, d^2y/dx^2=-6,<0 so max point (-1,2)
point of inflection at (0,0)
cubic graph, sketch points