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06:51
06:15 06:51if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.
d. Identify the local extreme values and inflection points of ƒ .
{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>
c. Find the time at which the object passes the rest position for the second time.
{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.
c. By experimentation, determine the approximate value of a (3 < a < 4)at which the graph separates into two curves.
Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.
c. For arbitrary positive values of K and b, when does the maximum growth rate occur (in terms of K and b)?
{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).
c. Graph g for a = 2, 3, and 4.
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. If f(x) = mx + b, then the linear approximation to f at any point is L(x) = f(x).
