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Section 3.1 Applications of Derivatives

This section is about attaching meaning to the mechanics -- new ideas, not new calculations.

Consider the following graph:

Figure 3.1.1.

If we look at this graph, we can describe it as increasing, then decreasing, then increasing again.

Definition 3.1.2.

Let \(f\) be defined on an interval containing \(x_1\) and \(x_2\text{:}\)

  • \(f\) is increasing on the interval if \(f(x_1) \lt f(x_2)\) whenever \(x_1 \lt x_2\)

  • \(f\) is decreasing if \(f(x_1) \gt f(x_2)\) whenever \(x_1 \gt x_2\)

  • \(f\) is constant if \(f(x_1) = f(x_2)\) for all \(x_1\) and \(x_2\)

When graphing functions, there are some important points that we need to be aware of:

  • critical points: \(f'(x) = 0\) or \(f'(x)\) is undefined

  • inflection points: inflection points only occur at places where \(f''(x) = 0\) and occurs where concavity changes

Note 3.1.10.

\(f''(x) = 0\) does not guarantee that there is an inflection point.

We will assign behavior on closed intervals and concavity on open intervals.

Example 3.1.1.

Use \(f'\) and \(f''\) to find intervals of increasing/decreasing and concavity for the function, \(f(x) = 2x^6 - 4x^3\text{:}\)

  • \(\displaystyle f'(x)\)

    \begin{align*} f'(x) \amp = \dfrac{d}{dx}(2x^6 - 4x^3)\\ \amp = 12x^5 - 12x^2\\ \amp \ldots\\ \amp = 12x^2(x - 1)(x^2 + x + 1) \end{align*}

  • \(\displaystyle f''(x)\)

    \begin{align*} f''(x) \amp = \dfrac{d}{dx}(12x^5 - 12x^2)\\ \amp = 60x^4 - 24x\\ \amp = 12x(5x^3 -2) \end{align*}

Find critical points, set \(f'(x) = 0\) and solve for \(x\text{:}\)

\begin{align*} 0 = \amp 12x^2 (x - 1)(x^2 + x + 1)\\ \amp x = 0, x = 1, \text{never zero} \end{align*}

Critical points are \(x = 0\) and \(x = 1\) (* both are stationary points).

Find the possible inflection points. Set \(f''(x) = 0\) and solve for \(x\text{:}\)

\begin{align*} 0 = \amp 12x (5x^2 - 2)\\ \amp x = 0, x = \sqrt[3]{\frac{2}{5}} \approx 0.736 \end{align*}

Analysis:

  • intervals of increasing/descreasing (2 critical points: \(x = 0\text{,}\) \(x = 1\))

    intervals \((12x^2)(x-1)(x^2+x+1)\) \(f'(x)\) conclusion
    \(x \lt 0\) \((+)(-)(+)\) \(- \) decreasing \((- \infty, 0]\)
    \(0 \lt x \lt 1\) \((+)(-)(+)\) \(- \) decreasing \([0,1]\)
    \(1 \lt x \) \((+)(+)(+) \) \(+ \) increasing \([1, \infty) \)

  • intervals of concavity (2 possible inflection points: \(x = 0\text{,}\) \(x = \sqrt[3]{\frac{2}{5}}\))

    intervals \((12x)(5x^3 - 2)\) \(f''(x)\) conclusion
    \(x \lt 0\) \((-)(-)\) \(+\) concave up \((\infty, 0)\)
    \(0 \lt x \lt \sqrt[3]{\frac{2}{5}}\) \((+)(-)\) \(-\) concave down \(( 0, \sqrt[3]{\frac{2}{5}})\)
    \(\sqrt[3]{\frac{2}{5}} \lt x\) \((+)(+)\) \(+\) concave up \(( \sqrt[3]{\frac{2}{5}}, \infty)\)

Finally, here is a graph of what the function looks like.

Figure 3.1.11. \(f(x) = 2x^6 - 4x^3\)