CALC HOMEWORK!! PLEASE HELP!! Determine where the function is concave upward and where it is concave downward. f(x) = 3x^4 - 6x^3 +x - 9?

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  • 3 years ago

    f(x) = 3x⁴ - 6x³ + x - 9

    f '(x) = 12x³ - 18x² + 1

    f ' '(x) = 36x² - 36x

    The graph of f ' '(x) is an upwards-opening parabola (because the coefficient of the x² term is positive). When f ' '(x) > 0, the original function will be concave up. If you find the x-values where f ' '(x) = 0, you'll know that f ' '(x) > 0 outside of those points.

    So here's f ' '(x) = 0 :

    36x² - 36x = 0

    x² - x = 0 ……………….. divided both sides by 36

    x(x - 1) = 0 …………….. factored left side

    By the principle of zero products, the solutions are x = 0 and x = 1.

    f ' '(x) > 0 outside the interval (0, 1), and so the original function is concave up on the intervals (- ∞, 0) and (1, ∞).

    The function is concave down when f ' '(x) < 0. In this case, it is concave down on (0, 1) .

    Here's a graph of the function. I put in the points at x = 0 and x = 1 so that it's easier to see that the function is concave down on (0, 1).

    https://www.desmos.com/calculator/4y1ymzgspq

    There's also a hidden graph there of the second derivative function f ' '(x) = 36x² - 36x . If you click on the currently hollow circle next to that function in the left panel, it will appear. You can see from that graph where f ' '(x) is greater than and less than zero, which agrees with what was said above.

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    Note: I realized later something I said isn't always true, namely, the part about "If you find the x-values where f ' '(x) = 0, you'll know that f ' '(x) > 0 outside of those points." That works if you have an upwards-opening parabola with vertex below the x-axis for the second derivative . However, if the second derivative is a quadratic, it's also possible that there might be only one point where f ' '(x) = 0, or even no points where f ' '(x) = 0. Yet, the method is valid. If the second derivative is a quadratic, you can work with the graph of that quadratic to figure out where it is greater or less than zero.

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