Question

Love this called sketch a graph of the polynomial function f(x)=x^3-6x^2+3x+10. Use it to complete the following:

Answer 1

Answer:

• increasing
• positive
• negative

Step-by-step explanation:

You can actually answer this question without graphing the equation, but a graph confirms the answers.

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A cubic with a positive leading coefficient will be negative and increasing on any interval* whose left end is -∞. Similarly, it will be positive and increasing on any interval whose right end is +∞.

The answer choices tell you ...

• there are zeros at -1, 2, 5
• there are turning points at 0.27, 3.73

The function is increasing up to the first turning point and after the second one.

The function is negative up to the first zero and between the last two.

• f is increasing on the intervals (-∞, 0.27) and (3.73, ∞).
• f is positive on the intervals (-1,2) and (5, ∞).
• f is negative on the intervals (-∞, -1) and (2,5).

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* We say "any interval" but we mean any interval whose boundary is a zero or turning point, and which properly describes an interval where the function is one of increasing, decreasing, positive, or negative.