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# Derivatives, Polynomials, Points of Inflection and Differential Equations

See the attached file.
4. Let h(x) be a function defined for all ... such that h(4) = ?3 and the derivative of h(x) is given by ....
(a) Find all values of x for which the graph of Ii has a horizontal tangent, arid determine whether 1 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 I concave up? Justify your answer.
(c) Write an equation for the line tangent to the graph of h(x) at x = 4.
(d) Does the line tangent to the graph of ...
5. A cubic polynomial function f is defined by
....
where a, b, and k are constants. The function f has a local minimum at x = ?L and the graph of f has a point of inflection at x z ?2.
(a) Find the values of a and b.
(h) If J f(x) (iX : 32, what is th 'alue of k 7
6. The function f is differentiable for all real numbers. slope at each point (x, v) on the graph is given by -a '6 2x).
(a) Find --_- and evaluate it al the point (3 --).
(b) Find y = f(x) by solving the differential equation = v2(6 - 2x) with the initial condition f(3) = .

Please see the attached file for the fully formatted problems.

#### Solution Summary

Derivatives, polynomials, points of inflection and differential equations are investigated and discussed in the solution.

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