Let's break down the two questions from the image.

Question 1: Decreasing Interval and Inflection Point

• The graph shows a function, and we need to find the interval where the function is decreasing and the location of the inflection point.

  1. A function is decreasing where the slope of the graph is negative. From the graph, the function appears to be decreasing between the peaks and troughs (where it slopes downwards).

     • The decreasing interval is likely between $x = 1$ and $x = 5$, but let me know if you'd like a closer look at specific points.

  2. The inflection point occurs where the concavity of the graph changes (from concave up to concave down or vice versa).

     • In the graph, this appears to occur near $x = 2$.

Tentative answers:

• The function is decreasing on the interval $1 < x < 5$.
• The inflection point is at $x = 2$.

Question 2: Finding Inflection Points and Concavity

You are given the function: $f(x) = 12x^5 + 30x^4 - 300x^3 + 7$

• Inflection Points: These occur where the second derivative changes sign.

  1. Find the first derivative: $f'(x) = 60x^4 + 120x^3 - 900x^2$
  2. Now, find the second derivative: $f''(x) = 240x^3 + 360x^2 - 1800x$
  3. Set $f''(x) = 0$ to find the inflection points: $240x^3 + 360x^2 - 1800x = 0$ Factor out the common term: $60x(4x^2 + 6x - 30) = 0$ This gives one solution at $x = 0$, and solving $4x^2 + 6x - 30 = 0$ will give the other two inflection points.

  Solving $4x^2 + 6x - 30 = 0$ using the quadratic formula: $x = \frac{-6 \pm \sqrt{6^2 - 4(4)(-30)}}{2(4)}$ $x = \frac{-6 \pm \sqrt{36 + 480}}{8} = \frac{-6 \pm \sqrt{516}}{8}$ Approximate the solutions: $x = \frac{-6 \pm 22.7}{8}$ $x_1 \approx 2.08, \quad x_2 \approx -3.58$ So, the inflection points are approximately $x = 0$, $x = 2.08$, and $x = -3.58$.

• Concavity Intervals: You can determine concavity by analyzing the sign of $f''(x)$ in the intervals formed by these inflection points:

  1. $(-\infty, -3.58)$
  2. $(-3.58, 0)$
  3. $(0, 2.08)$
  4. $(2.08, \infty)$

Would you like further details on solving the quadratic or determining concavity from these intervals?

Relative questions:

1. What is the difference between inflection points and critical points in a function?
2. How do you determine where a graph is concave up or concave down using derivatives?
3. Can a function have more than one inflection point, and what does that indicate about its graph?
4. How does the second derivative test help in finding concavity and inflection points?
5. What other methods can you use to determine where a function is increasing or decreasing?

Tip:

When solving for inflection points, remember that they only exist where the second derivative changes sign, not just where it equals zero!