Applied Calculus/Linear Algebra - Suppose that f is a function such that f'(x)=x^3+2x^2-7x+4 A) Explain why x=1 is a critical number (critical point) for f. B) Is it true that f has a local minimum value at x=1? Justify your claim. C) Is it true that f is concave down within some interval that contains x=2? Justify your claim.

Question

Applied Calculus/Linear Algebra - Suppose that f is a function such that f'(x)=x^3+2x^2-7x+4   A) Explain why x=1 is a critical number (critical point) for f. B) Is it true that f has a local minimum value at x=1? Justify your claim. C) Is it true that f is concave down within some interval that contains x=2? Justify your claim.

anonymous · Answer

1 is a critical point if 
$$f'(1)=0$$ so you can check that this is true

anonymous · Answer

I don't see the linear algebra party.

anonymous · Answer

$$1+2-7+4=0$$ is right, so you know 1 is a critical point

experimentx · Answer

it's because .. at 1, you have slope zero .. it means it's turning curve where you have either maxima or minima

anonymous · Answer

it could be a local max, local min, or neither.  find 
$$f''(x)=3x^2+4x-7$$ and see that 
$$f''(1)=0$$ as well

anonymous · Answer

in fact what this tells you is that 1 is a zero of f'(x) with multiplicity 2, so f' does not change sign at x =1, meaning this is not in fact a local min nor a local max

anonymous · Answer

1 is an inflection point of f, not a max or a min

anonymous · Answer

if f is concave down on some interval containing 2, that would mean that 
$$f''(2)<0$$ and you can check this