Calculus - Functions?

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#1. A cubic polynomial function f is defined by f(x) = 4x^3 +ax^2 + bx + k
where a, b and k are constants. The function f has a local minimum at x = -1, and the graph of f has a point of inflection at x= -2

a.) Find the values of a and b


#2. Let h be a function defined for all x (not equal to) 0, such that h(4) = -3 and the derivative of h is given by h'(x) = (x^2 - 2) / (x) for all x (not equal to) 0.

a.) Find all values of x for which the graph of h has a horizontal tangent, and determine whether h has a local maximum, a local minimum, or neither at each of these values. Justify your answers.

b.) On what intervals, if any, is the graph of h concave up? Justify

c.) Write an equation for the line tangent to the graph of h at x=4

d.) Does the line tangent to the graph of h at x = 4 lie above or below the graph of h for x > 4 ? Why?

  • Calculus - Functions? -

    I will do #1 for you.

    Show some work or steps that you have done for #2 and#4 and I will evaluate your work.

    #1:

    f '(x) = 12x^2 + 2ax + b
    f ''(x) = 24x + 2a

    f '(-1) = 0 = 12 - 2a +b ---> 2a - b = 12
    f ''(-2) = 0 = -48 + 2a ----> a= 24
    then 2(24) - b = 12
    b = 36

    a = 24 , b = 36

  • Calculus - Functions? -

    a billion

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