the graph of the derivative $f(x)$ is given below. on what interval(s) is the function $f(x)$ concave down…

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

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

Answer

Explanation:

Step1: Understand Concavity and Derivatives

A function $f(x)$ is concave down on an interval if its second derivative, $f''(x)$, is negative on that interval. Since $f''(x)$ is the derivative of $f'(x)$, this means that $f(x)$ is concave down where $f'(x)$ is decreasing.

Step2: Identify Intervals where $f'(x)$ is Decreasing

Observe the given graph of $f'(x)$. We need to find the x-intervals where the graph of $f'(x)$ has a negative slope (i.e., it is going downwards as we move from left to right). From the graph:

  • $f'(x)$ is decreasing from $x = -\infty$ until $x = -6$.
  • $f'(x)$ is increasing from $x = -6$ until $x = -4$.
  • $f'(x)$ is decreasing from $x = -4$ until $x = 0$.
  • $f'(x)$ is increasing from $x = 0$ until $x = \infty$.

Step3: Write the Intervals

The intervals where $f'(x)$ is decreasing are $(-\infty, -6)$ and $(-4, 0)$. These are the intervals where $f(x)$ is concave down.

Answer:

$(-\infty, -6), (-4, 0)$