Set $ h'(x) = 0 $ to find where the slope is horizontal: - Decision Point

Understanding the Context

The derivative $ h'(x) $ represents the rate of change (slope) of the function $ h(x) $ at any point $ x $. When $ h'(x) = 0 $, the function momentarily stops increasing or decreasing. In geometric terms, this means the tangent line at that point is horizontal.

Why is this important?

• It helps locate potential local extrema (maxima and minima), where the function changes direction.
  
• It aids in sketching accurate graphs by identifying key slope-shifting points.
  
• It supports real-world applications, such as finding optimal production levels or maximum profits.

How to Solve $ h'(x) = 0 $

1. Compute the Derivative:
   First, differentiate $ h(x) $ to find $ h'(x) $ exactly. This step requires proper application of differentiation rules—power, product, quotient, or chain rule—depending on $ h(x) $.

Key Insights

2. Set Derivative Equal to Zero:
   Solve the equation $ h'(x) = 0 $ for $ x $. This often involves factoring polynomial derivatives, simplifying expressions, or isolating $ x $ using algebraic methods.

3. Verify Real Solutions:
   Ensure the solutions are real numbers and within the domain of $ h(x) $. Complex or undefined points are excluded.

4. Analyze Critical Points:
   Use the First and Second Derivative Tests to classify the nature of each solution:
   

   • If $ h'(x) $ changes from positive to negative: local maximum.
     
   • If $ h'(x) $ changes from negative to positive: local minimum.
     
   • No sign change: may indicate a point of inflection or saddle point.

Practical Example

Consider $ h(x) = x^3 - 6x^2 + 9x + 1 $.
Step 1: Differentiate:
$$
h'(x) = 3x^2 - 12x + 9
$$
Step 2: Set $ h'(x) = 0 $:
$$
3x^2 - 12x + 9 = 0
$$
Divide by 3:
$$
x^2 - 4x + 3 = 0
$$
Factor:
$$
(x - 1)(x - 3) = 0
$$
Critical points: $ x = 1 $ and $ x = 3 $.

Final Thoughts

Step 3: Apply the Second Derivative Test:
$$
h''(x) = 6x - 12
$$

• At $ x = 1 $: $ h''(1) = -6 $ (concave down → local maximum).
  
• At $ x = 3 $: $ h''(3) = 6 $ (concave up → local minimum).

Summary

Solving $ h'(x) = 0 $ is central to calculus problem-solving. This equation reveals horizontal tangents, critical points, and key insights into function behavior. Whether graphing, optimization, or applied modeling, identifying where slopes are zero allows deeper analysis and precise understanding of dynamic systems.

Key Takeaways:

• $ h'(x) = 0 $ finds horizontal tangents and critical points.
  
• Always interpret results using derivative tests and function context.
  
• This method is foundational in optimization and calculus-based sciences.

Advanced Tip: Use computational tools or graphing calculators to verify solutions and visualize function behavior when working with complex derivatives. For polynomial or rational functions, symbolic algebra software can rapidly return analytical results, saving time and confirming manual calculations.