Question Type 1: Finding intervals where a function is increasing or decreasing Exercises

Find the derivative $f'(x)$ of the function $f(x) = x^3 - 5x^2 + 6x + 9$

Identify the critical points of $f(x) = x^3 - 5x^2 + 6x + 9$.

Solve the equation $f'(x)=0$ for $f(x) = x^3 - 5x^2 + 6x + 9$.

Determine the intervals on which $f(x) = x^3 - 5x^2 + 6x + 9$ is decreasing.

Construct a sign chart for $f'(x)$ of $f(x)=x^3-5x^2+6x+9$, indicating the sign of $f'(x)$ on each interval determined by the critical points.

Determine the intervals on which $f(x) = x^3 - 5x^2 + 6x + 9$ is increasing.

Determine whether $f(x)=x^3-5x^2+6x+9$ is strictly increasing, strictly decreasing, or neither on the interval $[0,2]$.

Show that $f(x)=x^3-5x^2+6x+9$ is not monotonic on its entire domain $\mathbb{R}$.

Classify each critical point of $f(x) = x^3 - 5x^2 + 6x + 9$ as a local maximum or minimum using the first derivative test.

Find the inflection point of $f(x)=x^3-5x^2+6x+9$ by using the second derivative.

Verify by selecting appropriate test points that $f(x)=x^3-5x^2+6x+9$ is increasing on $(-\infty,\tfrac{5-\sqrt7}{3})$ and decreasing on $(\tfrac{5-\sqrt7}{3},\tfrac{5+\sqrt7}{3})$.

Using the second derivative test, classify the critical points of $f(x)=x^3-5x^2+6x+9$.