Applications of Differentiation: Increasing/Decreasing Functions, Local Extrema, and Concavity

Applications of Differentiation

Increasing and Decreasing Functions

Understanding when a function is increasing or decreasing is fundamental in calculus. The behavior of a function is determined by the sign of its first derivative.

• Increasing Function: A function f(x) is increasing on an interval [a, b] if for all a < b in the domain, f(a) < f(b). This occurs when f'(x) > 0 on the interval.

• Decreasing Function: A function f(x) is decreasing on an interval [a, b] if for all a < b in the domain, f(a) > f(b). This occurs when f'(x) < 0 on the interval.

Example: Determine intervals of increase or decrease for f(x) = 2x3 - 9x2 + 12x.

• Compute the derivative: $f'(x) = 6x^2 - 18x + 12$

• Set f'(x) = 0 to find critical points: $6x^2 - 18x + 12 = 0 \Rightarrow x^2 - 3x + 2 = 0 \Rightarrow (x-2)(x-1) = 0$ So, $x = 1$ and $x = 2$ are critical points.

• Test intervals:

  • $x < 1$: $f'(x) > 0$ (increasing)

  • $1 < x < 2$: $f'(x) < 0$ (decreasing)

  • $x > 2$: $f'(x) > 0$ (increasing)

Critical Numbers and the First Derivative Test

Critical numbers are values of x where f'(x) = 0 or f'(x) is undefined, provided x is in the domain of f(x). The First Derivative Test helps classify these points as local maxima, minima, or neither.

• First Derivative Test:

  • If f'(x) changes from positive to negative at x = c, f has a local maximum at c.

  • If f'(x) changes from negative to positive at x = c, f has a local minimum at c.

  • If f'(x) does not change sign, f has no local extremum at c.

Example: Find local extreme values of f(x) = x4 + 4x3.

• Compute the derivative: $f'(x) = 4x^3 + 12x^2 = 4x^2(x+3)$

• Set f'(x) = 0: $4x^2(x+3) = 0 \Rightarrow x = 0, x = -3$

• Test intervals:

  • $x < -3$: $f'(x) < 0$ (decreasing)

  • $-3 < x < 0$: $f'(x) > 0$ (increasing)

  • $x > 0$: $f'(x) > 0$ (increasing)

• Conclusion:

  • At $x = -3$, $f'(x)$ changes from negative to positive: local minimum.

  • At $x = 0$, $f'(x)$ does not change sign: neither maximum nor minimum.

Intervals of Increase/Decrease for Rational Functions

For rational functions, critical points can also occur where the derivative is undefined, but only if those points are in the domain of the original function.

Example: Find intervals where f(x) = x2 / (x2 - 1) is increasing or decreasing.

• Compute the derivative: $f'(x) = \frac{-4x}{(x^2-1)^2}$

• Set f'(x) = 0: $-4x = 0 \Rightarrow x = 0$

• Points where f'(x) is undefined: $x = 1, x = -1$ (not in domain of f(x))

• Test intervals:

  • $x < -1$: $f'(x) > 0$ (increasing)

  • $-1 < x < 0$: $f'(x) < 0$ (decreasing)

  • $0 < x < 1$: $f'(x) > 0$ (increasing)

  • $x > 1$: $f'(x) > 0$ (increasing)

Concavity and Inflection Points

Concavity describes the direction in which a function curves. The second derivative f''(x) is used to determine concavity and locate inflection points.

• Concave Up: If f''(x) > 0 on an interval, f is concave upward there.

• Concave Down: If f''(x) < 0 on an interval, f is concave downward there.

• Inflection Point: A point where f''(x) = 0 and concavity changes.

Example: Find inflection points and intervals of concavity for f(x) = x4 - 4x3.

• Compute the second derivative: $f''(x) = 12x^2 - 24x$

• Set f''(x) = 0: $12x^2 - 24x = 0 \Rightarrow x(x-2) = 0$ So, $x = 0$ and $x = 2$ are possible inflection points.

• Test intervals:

  • $x < 0$: $f''(x) > 0$ (concave up)

  • $0 < x < 2$: $f''(x) < 0$ (concave down)

  • $x > 2$: $f''(x) > 0$ (concave up)

Summary Table: First and Second Derivative Tests

Test	Condition	Conclusion
First Derivative Test	$f'(c) = 0$ or undefined	Critical number
First Derivative Test	$f'(x)$ changes + to - at $c$	Local maximum at $c$
First Derivative Test	$f'(x)$ changes - to + at $c$	Local minimum at $c$
Second Derivative Test	$f''(c) > 0$	Concave up at $c$
Second Derivative Test	$f''(c) < 0$	Concave down at $c$
Inflection Point	$f''(c) = 0$ and concavity changes	Inflection point at $c$

Key Definitions

• Critical Number: A value c in the domain of f where f'(c) = 0 or f'(c) is undefined.

• Local Maximum/Minimum: The highest/lowest value of f in a neighborhood around a point.

• Inflection Point: A point where the graph changes concavity.

Additional info:

• All examples use polynomial and rational functions, but the principles apply to any differentiable function.

• Critical points where the derivative is undefined must be in the domain of the original function to be considered.