88. Given that x>0, find the maximum value, if any, of
a. x^(1/x)

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

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.

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.

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

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

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

$f\(\left\)(x\(\right\))=\(\frac{x^2-4}{x^2+1}\)$

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

$f\(\left\)(x\(\right\))=4\(\sin\) x\(\cos\) x$; $0$ < $x$ < $\(\pi\)$

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.

f(x) = 6x² - x³