Determine the open intervals on which the graph of the function is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)

Question

Determine the open intervals on which the graph of the function is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)

f(x) = x² − 2x + 3

Answer 1

Find the second derivative of the function:

f(x) = x²-2x+3

f'(x) = 2x-2

f''(x) = 2

Since f''(x)=2≠0, then there is no inflection point. Also, since f''(2)>0, this means that the graph is concave upward from (-∞,∞).

Hope this helped!

Answer 2

To determine the concavity of a function, we want to first find the second derivative of our function. From there, we see that a function if concave upward for x such that f''(x) > 0 and a function is concave downward for x such that f''(x) < 0.

Let's differentiate f(x)

f'(x) = 2x - 2 (by power rule and constant rule)

f''(x) = 2 (by power rule and constant rule)

As f''(x) = 2, we have f''(x) > 0 for all real numbers, hence f(x) is concave upward on (-∞, ∞).

Our graph is never concave downward.