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05:03{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).
d. Find the number and location of the fixed points of g for a = 2, 3, and 4 on the interval 0 ≤ x ≤ 1.
if 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 .
Rectangles in triangles Find the dimensions and area of the rectangle of maximum area that can be inscribed in the following figures.
d. An arbitrary triangle with a given area A (The result applies to any triangle, but first consider triangles for which all the angles are less than or equal to 90° .)
Sketch a graph of a function f with the following properties.
f' < 0 and f" < 0, for 8 < x < 10
{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).
