AP Calculous

let f be the function defined by f(x)=3X^5 -5X^3 +2
a) on what interval is f increasing? b) on what interval is the graph of f concave upward?
c)Write the equation of each horizontal line tangent to the graph of f

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  1. f is increasing when f' is positive

    f' = 15x^4 - 15x^2 = 15x^2 (x^2-1)
    So, f' > 0 when |x| > 1

    f is concave upward when f'' is positive

    f'' = 60x^3 - 30x = 30x(2x-1)
    So, f'' > 0 when x < 0 or x > 1/2

    Horizontal lines have slope=0. So, we want places where f'(x) = 0

    15x^2 (x^2 - 1) = 0
    x = -1, 0, 1
    The horizontal lines are
    y=f(-1)
    y=f(0)
    y=f(1)
    evaluate f(x) at those points to get your lines.

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  2. Oops. f'' = 30x(2x^2 - 1)
    so -1/√2 < x < 0 or x > 1/√2

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  3. a) That would be where the derivative
    f'(x) = 15x^4 -15x^2 > 0
    x^2*(x^2-1) >0
    Since x^2 must be positive or zero,
    (x+1)(x-1) > 0
    x > 1 or x<-1
    b) That would be where f"(x) > 0
    c) Horizontal tangents would be where f'(x) = 0.
    Find those x and y values.

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