Growth rate functions
a. Show that the logistic growth rate function f(P)=rP(1−P/K) has a maximum value of rK/4 at the point P=K/2.

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

4. y=9/14x^(1/3)(x^2-7)

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

5. y=x+sin(2x), -2π/3≤x≤2π/3

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

7. y=sin|x|, -2π≤x≤2π

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

8. y = 2cosx - √2x, -π≤x≤3π/2

155. Which is bigger, πᵉ or e^π?

Calculators have taken some of the mystery out of this once-challenging question.

(Go ahead and check; you will see that it is a very close call.)

You can answer the question without a calculator, though.

d. Conclude that

xᵉ < eˣfor all positivex ≠ e.

88. Given that x>0, find the maximum value, if any, of

a. x^(1/x)

Use the second derivative test to find the local extrema of the given function.

$g(x)=x^3-6x^2+9x+2$