Determine if the statement is equivalent to (means the same thing as) the statement
"f is concave down."
Explain your answer (whether the statement is equivalent or not).
f is decreasing.
Answer
"f is decreasing" does not mean the same thing as "f is concave down." We can have a function that is decreasing and concave up:
We can also have a function that is increasing and concave down:
Example 2
Determine if the statement is equivalent to (means the same thing as) the statement
"f is concave down."
Explain your answer (whether the statement is equivalent or not).
f is negative.
Answer
"f is negative" does not mean the same thing as "f is concave down." We can have a function that is negative and concave up:
We can also have a function that is positive and concave down:
Example 3
Determine if the statement is equivalent to (means the same thing as) the statement
"f is concave down."
Explain your answer (whether the statement is equivalent or not).
f ' is decreasing.
Answer
"f ' is decreasing" does mean the same thing as "f is concave down."
Example 4
Determine if the statement is equivalent to (means the same thing as) the statement
"f is concave down."
Explain your answer (whether the statement is equivalent or not).
f ' is negative.
Answer
"f ' is negative" does not mean the same thing as "f is concave down." We can have a function that is decreasing (f ' negative) and concave up:
We can also have a function that is concave down but increasing (f ' positive):
Example 5
Determine if the statement is equivalent to (means the same thing as) the statement
"f is concave down."
Explain your answer (whether the statement is equivalent or not).
f " is decreasing.
Answer
"f " is decreasing" does not mean the same thing as "f is concave down." If f " is decreasing but positive, then f will be concave up. The sign of f " determines the concavity of f, not whether f " is increasing or decreasing.
Example 6
Determine if the statement is equivalent to (means the same thing as) the statement
"f is concave down."
Explain your answer (whether the statement is equivalent or not).
f " is negative.
Answer
"f " is negative" does mean the same thing as "f is concave down."
