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Math Help - Calculus

  1. #1
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    Arrow Calculus

    The function f(x)=x^3+ax^2+bx has a local minimum at x=3 and a point of inflection at x=-1. Find the values of a and b.
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  2. #2
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    Quote Originally Posted by SoYah View Post
    The function f(x)=x^3+ax^2+bx has a local minimum at x=3 and a point of inflection at x=-1. Find the values of a and b.
    Differentiate it, and set the result equal to zero.

     f'(x) = 3x^2 + 2ax + b = 0

    But you know that this equation has two solutions, x = 3, and x =-1.

    Apply them

     3(3)^2 + 2a(3) + b = 0

     3(-1)^2 + 2a(-1) + b = 0

    2 equations, 2 unknowns. Solve simultaneously.
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  3. #3
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    Smile solution

    If f(x)= x^3+ax^2+bx, then
    f '(x)= 3x^2+2ax+b and
    f "(x)= 6x+2a.

    f(x) has an inflection point at x= -1, which means that f "(x)=0 when x=-1. So f "(x)= 6x+2a=0 when x=-1
    6(-1)+2a=0
    -6+2a=0
    2a=6
    a=3.

    f(x) also has a minimum at x=3, which means that f '(x)=0 when x=3. So,
    f '(x)= 3x^2+2ax+b=0 when x=3
    3(3)^2+2a(3)+b=0
    We already have the value for a, so we can substitute it as well.
    27+6(3)+b=0
    45+b=0
    b=-45.
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  4. #4
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    Quote Originally Posted by CaitSydney22 View Post
    If f(x)= x^3+ax^2+bx, then
    f '(x)= 3x^2+2ax+b and
    f "(x)= 6x+2a.

    f(x) has an inflection point at x= -1, which means that f "(x)=0 when x=-1. So f "(x)= 6x+2a=0 when x=-1
    6(-1)+2a=0
    -6+2a=0
    2a=6
    a=3.

    f(x) also has a minimum at x=3, which means that f '(x)=0 when x=3. So,
    f '(x)= 3x^2+2ax+b=0 when x=3
    3(3)^2+2a(3)+b=0
    We already have the value for a, so we can substitute it as well.
    27+6(3)+b=0
    45+b=0
    b=-45.
    It isn't really necessary to consider f''(x). See my post above.

    All you have to know is that the given x coordinates are the coordinates of stationary points, and hence are solutions to the equation f'(x) = 0. You have two of these points, and two unknown variables, so it is simply to formulate two equations which can be solved simultaneously to find the values of a and b.
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