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Question 994558: Find the interval [a, b] on which the function
f(x) = 2x^3 - 51x^2 + 420x + 3 is decreasing.
Enter your answer as: [a, b] for some constants a, b.
I am so confused! I was going to start by differentiating, but I'm not sure that is correct. Is this related to finding area of curves?
I am completely lost..
THANK YOU
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
In the first place, it is not possible for a continuous function to be increasing on a closed interval. If it increases, say, over its entire domain, then you have infinity on either end of your interval and such an interval is perforce open. If it changes somewhere from one direction to the other, it passes through a point where the derivative is zero and at that point, the function is neither increasing or decreasing and again, the interval is open at that point. Since you have a polynomial function, it is continuous and differentiable over its entire domain; namely, the real numbers.
The first thing you need to find are the critical numbers. These are values of the independent variable that make the first derivative either equal zero or be undefined.
In this case, the first derivative of this function is a polynomial which is defined for all real numbers. You will also find that the first derivative, which is a quadratic polynomial, has 2 real zeros.
You need to divide a number line into three segments, one from negative infinity to the smaller of the two zeros of the first derivative, the smaller zero to the larger zero, and the larger zero to infinity.
By testing values of the first derivative for non-endpoint values in those three intervals, you can determine the sign of the first derivative in those intervals. Where the sign is positive, the function is increasing and where the sign is negative, the function is decreasing.
John
My calculator said it, I believe it, that settles it
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