5.9 Connecting <latex>f'(x)</latex> and <latex>f''(x)</latex> with the Graph of <latex>f(x)</latex>

• When $f(x)$ is increasing, $f'(x)$ is positive
• Match the behavior of $f(x)$ with the sign of $f'(x)$:


  Increasing ↔️ Positive

  Decreasing ↔️ Negative

  Constant ↔️ Zero
• What does the derivative $f'(x)$ represent in calculus?


  Rate of change of $f(x)$
• What is the value of $f'(x)$ when $f(x)$ is constant?


  Zero
• If $f'(x) > 0$, the graph of $f(x)$ has an upward slope.


  True
• What is a point of inflection on the graph of $f(x)$?


  Where $f''(x)$ changes sign
• If $f'(x)$ is positive, $f(x)$ is increasing.


  True
• When $f'(x)$ is positive, f(x)</latex> is increasing.


  True
• Match the behavior of $f(x)$ with the sign of $f'(x)$:


  Increasing ↔️ Positive

  Decreasing ↔️ Negative

  Constant ↔️ Zero
• If $f(x)$ is increasing, $f'(x)$ is positive.


  True
• If $f''(x) < 0$, the graph of $f(x)$ is concave down.


  True
• When $f'(x) > 0$, $f''(x) > 0$, the graph of $f(x)$ is upward sloping and concave
• If $f'(x) > 0$ in an interval (a, b)</latex>, what is $f(x)$ doing in that interval?


  Increasing
• What is a critical point of $f(x)$?


  $f'(x) =$$0$ or undefined
• What does $f'(x)$ represent at a given point?


  Rate of change of $f(x)$
• If $f(x)$ is an increasing function, then $f'(x)$ is positive
• What is a point of inflection on the graph of $f(x)$?


  $f''(x) =$$0$ and changes sign
• If $f''(x)$ is positive, the graph of $f(x)$ is concave up.


  True
• If $f'(x)$ is negative, the graph of $f(x)$ is decreasing.


  True
• If $f'(x)$ changes from positive to negative at a critical point, it indicates a local maximum.


  True
• If f''(x) < 0</latex>, the graph of $f(x)$ is concave down.


  True
• If $f''(x)$ is positive, the concavity of f(x)</latex> is concave up
• For $f(x) =$$x^{3} - 6x^{2} +$$5$, what is $f''(x)$?


  $6x - 12$
• At $x =$$2$, $f''(x) =$$0$, indicating a point of inflection for f(x)</latex>.


  True
• Inflection points occur where $f''(x)$ is zero or undefined
• Steps to find inflection points for $f(x)$


  1️⃣ Find the second derivative $f''(x)$

  2️⃣ Set $f''(x) =$$0$

  3️⃣ Solve for $x$

  4️⃣ Analyze concavity around $x$

  5️⃣ Identify inflection points
• What sign change in $f''(x)$ indicates an inflection point?


  Positive to negative
• If $f'(x) < 0$, the graph of $f(x)$ is downward sloping.


  True
• The derivative $f'(x)$ represents the rate of change of the function $f(x)$ at a given point
• Summarize the relationship between f(x)</latex>, $f'(x)$, and the graph of $f(x)$:


  1️⃣ If $f'(x)$ is positive, $f(x)$ is increasing and the graph slopes upward.

  2️⃣ If $f'(x)$ is negative, $f(x)$ is decreasing and the graph slopes downward.

  3️⃣ If $f'(x)$ is zero, $f(x)$ is constant and the graph is a horizontal line.
• Match the second derivative $f''(x)$ with the concavity of $f(x)$:


  Positive ↔️ Concave up

  Negative ↔️ Concave down

  Zero ↔️ Possible point of inflection
• Arrange the steps to determine concavity and points of inflection using $f''(x)$:


  1️⃣ Find the second derivative $f''(x)$.

  2️⃣ Set $f''(x) =$$0$ to find possible points of inflection.

  3️⃣ Determine the sign of $f''(x)$ in intervals around critical points.

  4️⃣ Identify concavity as concave up if $f''(x) > 0$ and concave down if $f''(x) < 0$.

  5️⃣ If $f''(x)$ changes sign, it indicates a point of inflection.
• Match the derivative $f'(x)$ with the behavior of $f(x)$:


  Positive ↔️ Increasing

  Negative ↔️ Decreasing

  Zero ↔️ Constant
• If $f'(x)$ is zero at $x =$$c$, then $f(x)$ is momentarily constant
• Match the behavior of $f'(x)$ at $x =$$c$ with the type of extremum:


  Positive to negative ↔️ Local maximum

  Negative to positive ↔️ Local minimum

  No change in sign ↔️ Not an extremum
• If $f'(x)$ changes from negative to positive at x = c</latex>, then $f(x)$ has a local minimum at $x =$$c$.


  True
• Match the second derivative $f''(x)$ with the concavity of the graph of $f(x)$:


  Positive ↔️ Concave up

  Negative ↔️ Concave down

  Zero ↔️ Possible point of inflection
• If $f''(x) > 0$, the graph of $f(x)$ is concave up
• A point of inflection occurs when $f''(x) =$$0$ and the concavity of $f(x)$ changes
• What happens to the concavity of $f(x)$ at $x =$$2$ for $f(x) =$$x^{3} - 6x^{2} +$$5$?


  Changes concavity