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) = 6 − x − 8x⁴

^{Concave Downward = ?}

^{Concave upward= DNE}

Answer 1

First, we find possible inflection points by setting f''(x)=0:

f(x) = 6 − x − 8x⁴

f'(x) = -1 - 32x³

f''(x) = -96x²

0 = -96x²

x = 0

Now we can use the test points x = -1 and x = 1 to observe the function's concavity:

f''(-1) = -96(-1)² = -96 < 0

f''(1) = -96(1)² = -96 < 0

Hence, since we can see the function's second derivative is negative for both test points, this means that the open intervals on which the graph of the function is concave downward are (-∞,0)∪(0,∞).

Hope this helped!