Question

1. Show the calculus to find the critical numbers of f(x) = 6x² - x³. Then use (and show) the Second Derivative Test to determine whether they correspond to local maxima or local minima. 2. A function, f, has a domain of all real numbers. The 1st derivative is f'(x) = e?(x² + 2x + 1) and the 2nd derivative is f''(x) = e?(x² + 4x + 3). You must show all the tests and the value you are testing. Find the following, if they exist. If something doesn't exist, write “none”. The critical value(s) Interval(s) where f is concave upward Interval(s) where f is concave downward Type of relative extrema and the x-value where it occurs The x-value of any point(s) of inflection 3. Fill in the blanks and sketch the graph without a calculator. You must use calculus and show work. Find the y-intercept, give open intervals of concavity and list all point(s) of inflection and extrema as ordered pairs. Clearly label your axes and show the characteristics of the function on your graph. f(x) = x² / (x² - 4), f'(x) = -8x / (x² - 4)², f''(x) = 8(3x² + 4) / (x² - 4)³ Domain of f in interval notation y-intercept critical number(s) of f point(s) of inflection concave upward on concave downward on local extrema

          1. Show the calculus to find the critical numbers of f(x) = 6x² - x³. Then use (and show) the Second Derivative Test to determine whether they correspond to local maxima or local minima.

2. A function, f, has a domain of all real numbers.
The 1st derivative is f'(x) = e?(x² + 2x + 1) and the 2nd derivative is f''(x) = e?(x² + 4x + 3).
You must show all the tests and the value you are testing. Find the following, if they exist. If something doesn't exist, write “none”.

The critical value(s) ____________________

Interval(s) where f is concave upward ____________________

Interval(s) where f is concave downward ____________________

Type of relative extrema and
the x-value where it occurs ____________________

The x-value of any point(s) of inflection ____________________

3. Fill in the blanks and sketch the graph without a calculator. You must use calculus and show work. Find the y-intercept, give open intervals of concavity and list all point(s) of inflection and extrema as ordered pairs. Clearly label your axes and show the characteristics of the function on your graph.
f(x) = x² / (x² - 4), f'(x) = -8x / (x² - 4)², f''(x) = 8(3x² + 4) / (x² - 4)³

Domain of f in interval notation ____________________

y-intercept ____________________

critical number(s) of f ____________________

point(s) of inflection ____________________

concave upward on ____________________

concave downward on ____________________

local extrema ____________________

1. Show the calculus to find the critical numbers of f(x) = 6x² - x³. Then use (and show) the Second Derivative Test to determine whether they correspond to local maxima or local minima.

2. A function, f, has a domain of all real numbers.
The 1st derivative is f'(x) = e?(x² + 2x + 1) and the 2nd derivative is f”(x) = e?(x² + 4x + 3).
You must show all the tests and the value you are testing. Find the following, if they exist. If something doesn't exist, write “none”.

The critical value(s)

Interval(s) where f is concave upward

Interval(s) where f is concave downward

Type of relative extrema and
the x-value where it occurs

The x-value of any point(s) of inflection

3. Fill in the blanks and sketch the graph without a calculator. You must use calculus and show work. Find the y-intercept, give open intervals of concavity and list all point(s) of inflection and extrema as ordered pairs. Clearly label your axes and show the characteristics of the function on your graph.
f(x) = x² / (x² - 4), f'(x) = -8x / (x² - 4)², f”(x) = 8(3x² + 4) / (x² - 4)³

Domain of f in interval notation

y-intercept

critical number(s) of f

point(s) of inflection

concave upward on

concave downward on

local extrema

Added by Courtney G.

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