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