Find the maximum or minimum of: f(x) = x2 + 4x + 4. What is the value of x that produces the maximum (or minimum)

Question

anonymous · Answer

f(x)=x^2+16x

sburchette · Answer

By examining this function, we can see that it will have a minimum because a positive x^2 term will open upward. A min will occur where the slope of the tangent line to the curve is 0, so we can differentiate f(x) to obtain
$$f'(x)=2x+4$$ Setting this equal to zero and solving we get$$x=-2$$ So the function has a critical number at x=-2 and likely an extremum there as well.  We can take the derivative at t value less than the critical number and at a value greater than the critical number to determine whether the graph is increasing or decreasing. $$f'(-3)=-2$$$$f'(0)=4$$ So the function is decreasing to the left of the critical number and decreasing to the right. From this we can conclude that there is a minimum at the point (-2,0)

sburchette · Answer

Correction: I meant to say that the function is increasing to the right of the critical number