Solving F''(x) = 0 A Calculus Problem Explained
In this article, we will delve into the process of finding the solution to the equation f''(x) = 0, where f(x) is defined as 12x² - 4x + 7 + ln(x). This involves calculating the second derivative of the function and then solving the resulting equation. Such problems are common in calculus, particularly in optimization problems where we need to find points of inflection. Let's embark on this mathematical journey step by step.
Understanding the Function
Before we dive into the derivatives, let's take a moment to understand our function: f(x) = 12x² - 4x + 7 + ln(x). This function is a combination of a quadratic polynomial (12x² - 4x + 7) and a natural logarithm (ln(x)). The domain of the natural logarithm is x > 0, so we must keep this in mind as we work through the problem. The domain restriction is crucial because it will affect the possible solutions we can consider.
Step 1: Finding the First Derivative, f'(x)
To find the second derivative, we first need to compute the first derivative, f'(x). We will apply the power rule and the derivative of the natural logarithm. The power rule states that the derivative of x^n is nx^(n-1), and the derivative of ln(x) is 1/x. Applying these rules, we get:
f'(x) = d/dx [12x² - 4x + 7 + ln(x)]
f'(x) = 12 * d/dx [x²] - 4 * d/dx [x] + d/dx [7] + d/dx [ln(x)]
f'(x) = 12 * 2x - 4 * 1 + 0 + 1/x
f'(x) = 24x - 4 + 1/x
So, the first derivative f'(x) is 24x - 4 + 1/x. This derivative represents the rate of change of the original function f(x). Understanding the first derivative is essential because it helps us find critical points where the function may have local maxima or minima.
Step 2: Finding the Second Derivative, f''(x)
Now that we have the first derivative, we can find the second derivative, f''(x). The second derivative will give us information about the concavity of the original function. We differentiate f'(x) with respect to x:
f''(x) = d/dx [24x - 4 + 1/x]
f''(x) = d/dx [24x] - d/dx [4] + d/dx [x^(-1)]
f''(x) = 24 - 0 + (-1)x^(-2)
f''(x) = 24 - 1/x²
Thus, the second derivative f''(x) is 24 - 1/x². The second derivative is critical for identifying inflection points, which are points where the concavity of the function changes.
Step 3: Solving f''(x) = 0
Our goal is to solve the equation f''(x) = 0 for x. This will give us the possible x-values where the concavity of f(x) changes. We set our expression for f''(x) equal to zero and solve for x:
24 - 1/x² = 0
Add 1/x² to both sides:
24 = 1/x²
Multiply both sides by x²:
24x² = 1
Divide both sides by 24:
x² = 1/24
Take the square root of both sides:
x = ±√(1/24)
x = ±1/√(24)
Simplify the square root in the denominator:
x = ±1/(2√6)
Rationalize the denominator by multiplying the numerator and denominator by √6:
x = ±√6 / (2√6 * √6)
x = ±√6 / (2 * 6)
x = ±√6 / 12
So, we have two potential solutions: x = √6 / 12 and x = -√6 / 12. However, we must remember the domain restriction from the original function, ln(x), which requires x > 0. Therefore, we can discard the negative solution. This domain restriction significantly impacts our final solution.
Step 4: Considering the Domain Restriction
Recall that the domain of f(x) includes the natural logarithm ln(x), which is only defined for x > 0. Therefore, we must discard the negative solution x = -√6 / 12 since it is not in the domain of the original function. This leaves us with only one possible solution:
x = √6 / 12
Now, we need to approximate this value to the nearest thousandth:
x ≈ √6 / 12 ≈ 2.449 / 12 ≈ 0.204
Therefore, the solution to f''(x) = 0, rounded to the nearest thousandth, is x ≈ 0.204. The approximation is crucial for practical applications where an exact value may not be necessary or easily usable.
Step 5: Final Answer
In conclusion, by calculating the first and second derivatives of the given function f(x) = 12x² - 4x + 7 + ln(x) and solving the equation f''(x) = 0, we found that the only valid solution is approximately x ≈ 0.204. This solution respects the domain restriction imposed by the natural logarithm in the original function. The final answer is a result of careful differentiation, algebraic manipulation, and consideration of domain restrictions.
- Find the first derivative, f'(x): Applying the power rule and the derivative of ln(x), we found f'(x) = 24x - 4 + 1/x.
- Find the second derivative, f''(x): Differentiating f'(x), we obtained f''(x) = 24 - 1/x².
- Solve f''(x) = 0: Setting the second derivative to zero, we found x = ±√6 / 12.
- Consider the domain restriction: Since the domain of ln(x) is x > 0, we discarded the negative solution.
- Approximate the solution: We approximated x = √6 / 12 to x ≈ 0.204 to the nearest thousandth.
Conclusion
Solving equations involving derivatives is a fundamental skill in calculus. In this article, we methodically worked through the steps of finding the second derivative of the given function and solving for the points of inflection. By understanding the properties of derivatives and considering domain restrictions, we arrived at the correct solution. This process is not only important for mathematical exercises but also for real-world applications where optimization and analysis of functions are crucial. The thorough analysis and step-by-step solution provided here should serve as a valuable reference for anyone tackling similar problems in calculus.
By understanding each step—from finding derivatives to considering domain restrictions—one can confidently approach complex calculus problems. This systematic approach ensures accuracy and a deep comprehension of the underlying mathematical principles.
