5. Graphical Applications of Derivatives

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) = x²e⁻ˣ

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.

Each of Exercises 43–48 gives the first derivative of a function y = ƒ(𝓍). (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.

y' = 𝓍⁴ ― 2𝓍² 

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

Identifying Extrema

In Exercises 19–40:

b. Identify the function’s local extreme values, if any, saying where they occur.

f(θ) = 3θ² − 4θ³

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

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

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) = x³ - 13x² - 9x

{Use of Tech} Graph carefully Graph the function f(x) = 60x⁵ - 901x³ + 27x in the window [-4,4] x [-10,000, 10,000]. How many extreme values do you see? Locate all the extreme values by analyzing f'. 

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) = 2x² ln x - 11x²

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) = 2x³ - 3x² + 12

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

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.

p(t) = 2t³ + 3t² - 36t

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.

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) = 2x⁻³ - x⁻²

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

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