Use the first derivative to find all critical points and use the second derivati

Question

Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither.

Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither. f(x)=x^5-15x^4+55 Enter the exact answers in increasing order. If there is no answer, enter "none". The critical point at is local max, min or neither? The critical point at is local max, min, or neither? The inflection point is

Explanation / Answer

f'(x) = 5x^4 - 60x^3 = 5x^3(x-12) -> two crtiical points : x = 0 , x = 12

The critical point at x = 0 is local max.

The critical point at x = 12 is local min.

f"(x) = 20x^3 - 180x^2 = 20x^2 (x - 9) = 0 -> x = 9

f(9) = 9^5 - 15*9^4 + 55 = -39311

So the inflection point is (9 , -39311)

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