Ｃａｌｃｕｌｕｓ　１　ｌｅｃｔｕｒｅ　３．４　Tｈｅ　Sｅｃｏｎｄ　Dｅｒｉｖａｔｉｖｅ　Tｅｓｔ　Fｏｒ　Cｏｎｃａｖｉｔｙ　Oｆ　Fｕｎｃｔｉｏｎｓ

Ｉｎｔｒｏｄｕｃｔｉｏｎ

• [0:16]. The second derivative describes how a function’s slope changes, determining whether the curve is concave up or concave down. – [📷image]
  • If the second derivative is positive, the slope is increasing, and the curve is concave up.
  • If the second derivative is negative, the slope is decreasing, and the curve is concave down.
  • If $𝒇′′(𝓬)=0$, this is only a possible inflection point (PIP).
    • A brief note: to confirm whether an inflection point truly exists, it is necessary to check whether the sign of the second derivative changes on either side of that point.
    • If the sign does not change, then there is no real inflection.

Ｓｅｃｏｎｄ　ｄｅｒｉｖａｔｉｖｅ　ｔｅｓｔ

• [3:21]. The second derivative test is used to find inflection points. The steps are: – [📷image]
  • ❶ Find the function’s second derivative.
  • ❷ Set the second derivative equal to zero and solve for $𝒙$. This provides the possible inflection points.
    • Additionally, check for points where the second derivative does not exist (due to discontinuities, vertical asymptotes, etc.). These points can also be potential inflection points if the concavity changes.
  • ❸ Create a second derivative chart:
    • Place the possible inflection points on a number line.
    • Evaluate the sign of the second derivative in each interval.
    • A positive sign indicates concave up; a negative sign indicates concave down.
  • ❹ Remember that if $𝒇′′(𝓬)=0$, it is only a possible inflection point. To confirm if there is a real change in concavity, you must check the sign of $𝒇′′(𝒙)$ before and after $𝒙=𝓬$. If there is no change in sign, then there is no true inflection.
• [7:21]. The first derivative provides information about relative maxima and minima, while the second derivative provides information about concavity and inflection points.
• [8:00]. 🧩 Example –: Find the inflection points of the function $𝒇(𝒙)=𝒙^4-4𝒙^3+12$ – [📷image]
  • ❶ Find the first and second derivatives:
    • $𝒇′(𝒙)=4𝒙^3-12𝒙^2$
    • $𝒇′′(𝒙)=12𝒙^2-24𝒙$
  • ❷ Set the second derivative equal to zero and solve for $𝒙$:
    • $12𝒙^2-24𝒙=0$
    • $𝒙=0,\; 𝒙=2$
  • ❸ Create the second derivative chart:
  • ❹ The inflection points are $(0,12)$ and $(2,-4)$.
• [15:27]. 🧩 Example –: Find the inflection points of the function $𝓰(𝒙)=(𝒙-1)^{1/3}$ – [📷image]
  • ❶ Find the first and second derivatives:
    • $𝓰′(𝒙)=\dfrac{1}{3}(𝒙-1)^{-2/3}$
    • $𝓰′′(𝒙)= -\dfrac{2}{9}(𝒙-1)^{-5/3}$
  • ❷ Identify points where the second derivative is undefined:
    • $𝒙=1$
  • ❸ Create the second derivative chart:
  • ❹ The inflection point is $(1,0)$.

Ｃｏｍｂｉｎｉｎｇ　ｔｈｅ　ｆｉｒｓｔ　ａｎｄ　ｓｅｃｏｎｄ　ｄｅｒｉｖａｔｉｖｅ　ｔｅｓｔｓ

• [24:06]. The first and second derivative tests can be combined to get a complete picture of the function’s behavior.
• [25:10]. 🧩 Example –: Analyze the function $𝒉(𝒙)=𝒙^3-3𝒙^2+24𝒙-32$ – [📷image]
  • First Derivative Test:
    • $𝒉′(𝒙)=3𝒙^2-6𝒙+24$
    • Critical numbers: $𝒙=-2,\; 𝒙=4$
    • First derivative chart:
  • [29:29]. Second Derivative Test:
    • $𝒉′′(𝒙)=6𝒙-6$
    • Possible inflection point: $𝒙=1$
    • Second derivative chart:
  • Remember, the second derivative can also help classify critical points from the first derivative test:
    • $𝒇′(𝓬)=0$ and $𝒇′′(𝓬)>0$ → relative minimum at $𝒙=𝓬$.
    • $𝒇′(𝓬)=0$ and $𝒇′′(𝓬)<0$ → relative maximum at $𝒙=𝓬$.
    • $𝒇′(𝓬)=0$ and $𝒇′′(𝓬)=0$ → **inconclusive**, further analysis required.
  • [31:53]. Combining both tests provides the following information:
    • Relative maximum: $(-2,60)$
    • Relative minimum: $(4,-48)$
    • Inflection point: $(1,6)$

Ｌｉｍｉｔｓ　ａｔ　ｉｎｆｉｎｉｔｙ

• [36:01]. Section 3.5 of the course will address limits at infinity, allowing us to analyze a function’s behavior as $𝒙$ approaches positive or negative infinity.