Monotonicity Based Problems - Inequality Based Using Second Derivative

Monotonicity in mathematics refers to the behavior of a function as either non-increasing or non-decreasing when its domain is ordered. When dealing with inequalities, understanding the monotonicity of a function can be crucial for solving problems. One of the ways to analyze monotonicity is by using the second derivative of a function.

Understanding Monotonicity

A function $f(x)$ is said to be:

• Monotonically increasing on an interval if for any two numbers $a$ and $b$ in that interval, $a < b$ implies $f(a) \leq f(b)$.
• Strictly monotonically increasing if $a < b$ implies $f(a) < f(b)$.
• Monotonically decreasing on an interval if for any two numbers $a$ and $b$ in that interval, $a < b$ implies $f(a) \geq f(b)$.
• Strictly monotonically decreasing if $a < b$ implies $f(a) > f(b)$.

Second Derivative Test for Monotonicity

The second derivative of a function provides information about the concavity of the function. If the second derivative is positive on an interval, the function is concave up, and if it is negative, the function is concave down.

Second Derivative ($f''(x)$)	Concavity	Monotonicity of $f'(x)$
$f''(x) > 0$	Concave up	Increasing
$f''(x) < 0$	Concave down	Decreasing
$f''(x) = 0$	Inflection point	No conclusion

Using Second Derivative to Solve Inequalities

To solve inequality-based problems using the second derivative, follow these steps:

1. Find the first derivative ($f'(x)$): This represents the slope of the tangent line to the function at any point $x$.
2. Find the second derivative ($f''(x)$): This tells us about the concavity of the function.
3. Determine intervals of concavity: Use the sign of $f''(x)$ to determine where the function is concave up or down.
4. Analyze monotonicity: Since the first derivative is monotonically increasing where $f''(x) > 0$ and decreasing where $f''(x) < 0$, use this information to determine where $f(x)$ is increasing or decreasing.
5. Solve the inequality: Use the intervals of increase and decrease to solve the inequality.

Example

Let's consider the function $f(x) = x^3 - 3x^2 - 9x + 5$ and solve the inequality $f(x) > 0$.

1. First derivative: $f'(x) = 3x^2 - 6x - 9$
2. Second derivative: $f''(x) = 6x - 6$
3. Determine intervals of concavity:
   • Set $f''(x) = 0$: $6x - 6 = 0 \Rightarrow x = 1$
   • Test intervals around $x = 1$: For $x < 1$, $f''(x) < 0$ (concave down); for $x > 1$, $f''(x) > 0$ (concave up).
4. Analyze monotonicity:
   • Since $f''(x) > 0$ for $x > 1$, $f'(x)$ is increasing in this interval.
   • Since $f''(x) < 0$ for $x < 1$, $f'(x)$ is decreasing in this interval.
5. Solve the inequality:
   • Find the critical points of $f(x)$ by setting $f'(x) = 0$: $3x^2 - 6x - 9 = 0$.
   • Factor the quadratic to find the roots: $x = -1$ and $x = 3$.
   • Test intervals around the critical points to determine where $f(x) > 0$.

By analyzing the sign of $f(x)$ in each interval, we can determine where the function is above the x-axis (where $f(x) > 0$). In this example, $f(x) > 0$ for $x < -1$ and $x > 3$.

Conclusion

Using the second derivative to analyze monotonicity is a powerful tool for solving inequality-based problems. It allows us to determine where a function is increasing or decreasing, which is essential for understanding the behavior of the function and solving inequalities. Remember that the second derivative gives us information about the concavity of the function, which in turn helps us to deduce the monotonicity of its first derivative.