the graph of $f(x)$ is given below. on what interval(s) is the second derivative $f(x)$ positive? give your…

the graph of $f(x)$ is given below. on what interval(s) is the second derivative $f(x)$ positive? give your answer in interval notation, and use commas to separate multiple intervals if necessary.

the graph of $f(x)$ is given below. on what interval(s) is the second derivative $f(x)$ positive? give your answer in interval notation, and use commas to separate multiple intervals if necessary.

Answer

Explanation:

Step1: Identify Concavity

The second derivative $f''(x)$ is positive where the function $f(x)$ is concave upward. We need to identify these intervals from the given graph of $f(x)$.

Step2: Determine Intervals of Upward Concavity

Observe the graph to find where it curves upwards (like a "cup" opening upwards). From the graph, we can see:

  • The function is concave upward from the vertical asymptote at $x = -4$ to an inflection point at approximately $x = -2$.
  • The function is concave upward from an inflection point at approximately $x = 1$ to the vertical asymptote at $x = 3$.

Step3: Write the Intervals

The intervals where $f''(x) > 0$ are where $f(x)$ is concave upward. Based on the visual inspection of the graph, these intervals are $(-4, -2)$ and $(1, 3)$.

Answer:

$(-4, -2), (1, 3)$