Applied Calculus - 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 - 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.

dumbcow · Answer

critical points occur when slope is zero or f'(x) = 0
by plugging in x=1, you find f'(1) = 0

Find 2nd derivative to determine concavity and whether x=1 is min or max
f''(x) = 3x^2 +4x -7
f''(1) = 0
f''(2) = 13 --> positive which means concave up
so at x=1 you have an inflection point, f(x) transitioned from being concave down to concave up