BackIdentify 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
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) = eˣ(x - 2)²
Identifying Extrema
In Exercises 19–40:
b. Identify the function’s local extreme values, if any, saying where they occur.
f(r) = 3r³ + 16r
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = x(x − 1)
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
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' = 𝓍² ― 𝓍―6
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.
; < <
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)
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 - (3/2)x² - 36x
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = 1− 4/x², x ≠ 0
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.
2. y=x^4/4-2x^2+4
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = (x − 1)(x + 2)(x − 3)