f(x)=(x+2)^3+(x-1)^4
A critical number (or critical point) of a function is a point a within its domain at which f'(a)=0 or at which f'(a) is undefined, i.e. where f(a) is not differentiable at a.
First, find the domain of f(x).
Since it is a polynomial, its domain is (-∞∞), or ℝ.
It is also differentiable throughout its domain.
What we need to find is therefore the points at which f'(x)=0.
For
f(x)=(x+2)^3+(x-1)^4
we find using the standard rules of differentiation:
f'(x)=3(x+2)²+4(x-1)³
Expand to get:
f'(x)=4x^3-9x^2+24x+8
By graph plotting, or by trial and error, there is a zero around x=-0.3.
To make sure that there is only one zero, we calculate
f"(x)=12x^2-18x+24
which stays above the x-axis throughout its domain. This means that f'(x) is monotonically increasing, and hence only one zero can exist.
Now all we need to do is to refine the accuracy of the zero of f'(x) near x=-0.3 to be the only critical point.