Determine where the graph of $f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 6$ is concave up. A)Everywhere B)For x 3 C)Nowhere D)For -3

Question

Determine where the graph of $f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 6$ is concave up.
A)Everywhere
B)For x < -3 and x > 3
C)Nowhere
D)For -3 < x < 3

Quizplus · Accepted Answer

To determine where the graph of the function is concave up, we need to find the second derivative of $f(x)$ and analyze its sign. The first derivative $f'(x) = 4x^3 - 108x$, and the second derivative $f''(x) = 12x^2 - 108$. Setting $f''(x) > 0$ to find where the graph is concave up gives $12x^2 - 108 > 0$, which simplifies to $x^2 > 9$, or $x < -3$ and $x > 3$.

Determine Where the Graph Of f(x)=x4−54x2+6f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 6f(x)=x4−54x2+6

Determine Where the Graph Of $f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 6$