Find a example of a function g(x) ?

Question

Find a example of a function g(x) such that g'(3)=0 and g''(3)=0, but the function has an absolute minimum at x=3. Explain

Answer 1

g''(3)=0

g''(x)=?(x-3)²

g'(x)=?(1/3)(x-3)³

g(x)=?(1/12)(x-3)^4

sry all I can say is test it

I started with g''(x)=?(x-3)

but that didn't work out so

I just increased the power

Answer 2

eric has some point

the function is of the form:

g(x) = k(x-3)^4 , provided k > 0

since g'(x) = 4k (x-3)^3

g''(x) = 12k (x-3)^2

thus g'(3) = g''(3) = 0 .. . .

meanwhile to show that it has a minimum at 3, you can use first derivative test:

g'(x) < 0 , x < 3

g'(x) > 0 , x > 3

thus there is a relative minimum.

to show that it is absolute, you need the graph.

the graph of this quartic is almost parabolic.

thus the point (3,0) is an absolute minimum.