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Section 3.2 Relative Extrema

Definition 3.2.1.

Refers to minimum or maximum values as "optimum".

There are two techniques for interpretation:

  • FDT (First Derivative Test)

    Does the sign of \(f'\) change as we pass through a critical point? If so, how?

    • If \(f'\) switches from negative to positive, we say \(f\) has a relative minimum at its critical point.

    • if \(f'\) switches from positive to negative, we say \(f\) has a relative maximum at its critical point.

    If the derivative doesn't change sign there is no relative extreme value.

    Note 3.2.2.

    In order to use this method, we need the first derivative and two values/signs of the first derivative.

  • SDT (Second Derivative Test)

    How is the graph concavity a the critical point?

    • If \(f''(x) \gt 0\) at the critical point \(x_0\text{,}\) we say \(x_0\) is the local minimum.

    • If \(f''(x) \lt 0\) at the critical point \(x_0\text{,}\) we say \(x_0\) is the local maximum.

    • If \(f''(x) = 0\) the test is inconclusive and we look at FDT.

    Note 3.2.3.

    In order to use SDT, \(f\) must be twice differentiable at the stationary pont \(x_0\text{.}\)

Example 3.2.1.

Find the relative extreme by FDT / SDT.

Consider \(f(x) = x^3 - x^2 - x + 1 \text{:}\)

\begin{align*} f'(x) \amp = 3x^2 - 2x -1\\ f''(x) \amp = 6x - 2 = 2(3x - 1)\\ \amp (\text{critical points at } x = -\dfrac{1}{3}, 1 ) \end{align*}
Critical Points \(2(3x-1)\) \(f''(x)\) Result
\(x = -\dfrac{1}{3}\) \((+)(-)\) \((-)\) relative max at \(x = - \dfrac{1}{3}\text{;}\) value is \(f(- \dfrac{1}{3}) \approx 1.18\)
\(x = 1\) \((+)(+)\) \((+)\) relative min at \(x = 1 \text{;}\) value is \(f(1) = 0\)