5. Graphical Applications of Derivatives

109. Suppose the derivative of the function y = f(x) is

y'=(x-1)^2(x-2).

At what points, if any, does the graph of f have a local minimum, local maximum, or

point of inflection? (Hint: Draw the sign pattern for y'.)

110. Suppose the derivative of the function y = f(x) is

y'=(x-1)^22(x-2)(x-4).

At what points, if any, does the graph of f have a local minimum, local maximum, or

point of inflection?

119. Find the values of constants a, b, and c such that the graph of y = ax^3 + bx^2 + cx has a

local maximum at x = 3, local minimum at x =- 1, and inflection point at (1, 11).

Finding Extrema from Graphs

In Exercises 11–14, match the table with a graph.

Find values of a and b such that the function

ƒ(𝓍) = (a𝓍 + b) / 𝓍² ―1)

has a local extreme value of 1 at 𝓍 = 3. Is this extreme value a local maximum or a local minimum? Give reasons for your answer.

Finding Extrema from Graphs

In Exercises 11–14, match the table with a graph.

Finding Extreme Values

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

y = 𝓍³ + 𝓍² ― 8𝓍 + 5

101. In Exercises 101 and 102, the graph of f' is given. Determine x-values corresponding to local minima, local maxima, and inflection points for the graph of f.