Which graph represents the function f(x)=x4+4x3−12x2−32x+64?

The degree of f(x) is 4. Also the leading coefficient is 1 and it is positive

So  as x approaches infinity then y approaches infinity

as x approaches -infinity then y approaches infinity

The first and fourth graph goes up and it satisfies the above . so we ignore the second and third graph.

Now we check the x intercepts of the first  graph

x intercepts of first graph is -4  and 2

Plug in -4 for x in f(x) and check whether we get 0

Now plug in 2 for x and check

So -4  and 2  are the x intercepts that satisfies f(x)

Hence first option is the graph of

The function

has a derivative

The derivative of this function is zero when

The derivative function is negative on the left side of -4 and positive on the right side, it is also negative on the right hand side of -1, and positive on the right hand side of positive 2. From this information we can gather that the graph of f(x)  has a minima at x=-4, a maxima at x=-1 and a minima at x=-2.

The only graph that fits this description from the given ones is the first graph

the perimeter is increased by a factor of 2