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1 Increasing and Decreasing functions

1.1 Definition 4.3.1

Let \(f\) be a function and \(I\) an interval contained in the domain of \(f\).

  • We say that \(f\) is strictly increasing on \(I\) if for all numbers \(x_1, x_2 \in I\), if \(x_1 < x_2\) then \(f(x_1) < f(x_2)\)
  • We say that \(f\) is weakly increasing on \(I\) if for all numbers \(x_1, x_2 \in I\), if \(x_1 < x_2\) then \(f(x_1) \leq f(x_2)\)
  • We say that \(f\) is strictly decreasing on \(I\) if for all numbers \(x_1, x_2 \in I\), if \(x_1 < x_2\) then \(f(x_1) > f(x_2)\)
  • We say that \(f\) is weakly decreasing on \(I\) if for all numbers \(x_1, x_2 \in I\), if \(x_1 < x_2\) then \(f(x_1) \geq f(x_2)\)
  • We say that \(f\) is constant on \(I\) if for all numbers \(x_1, x_2 \in I\), \(f(x_1) = f(x_2)\)

1.2 Theorem 4.3.2

Suppose \(f\) is continous on an interval \(I\) and differentiable on the interior of \(I\).

  • If \(f'(x) > 0\) for all \(x\) in the interior of \(I\), then \(f\) is strictly increasing on \(I\).
  • If \(f'(x) \geq 0\) for all \(x\) in the interior of \(I\), then \(f\) is weakly increasing on \(I\).
  • If \(f'(x) < 0\) for all \(x\) in the interior of \(I\), then \(f\) is strictly decreasing on \(I\).
  • If \(f'(x) \leq 0\) for all \(x\) in the interior of \(I\), then \(f\) is weakly decreasing on \(I\).
  • If \(f'(x) = 0\) for all \(x\) in the interior of \(I\), then \(f\) is constant on \(I\).

1.3 Definition 4.3.3

Suppose that \(f\) is a function and \(a\) is a number in the domain of \(f\).

  • Suppose that for every \(x\) in the domain of \(f\), \(f(a) \leq f(x)\). Then we say that \(f(a)\) is the minimum value of \(f\). We also say that \(f\) attains its minimum value at \(a\).
  • Suppose that for every \(x\) in the domain of \(f\), \(f(a) \geq f(x)\). Then we say that \(f(a)\) is the maximum value of \(f\). We also say that \(f\) attains its maximum value at \(a\).

Additional points:

  • Not every function has maximum and miniumum values.

1.4 Definition 4.3.4

Suppose \(f\) is a function, \(S\) is a subset of the domain of \(f\), and \(a \in S\).

  • Suppose that for every \(x \in S\), \(f(a) \leq f(x)\). Then we say that \(f(a)\) is the minimum value of \(f\) on \(S\). We also say that \(f\) attains its minimum value on the set \(S\) at \(a\).
  • Suppose that for every \(x \in S\), \(f(a) \geq f(x)\). Then we say that \(f(a)\) is the maximum value of \(f\) on \(S\). We also say that \(f\) attains its maximum value on the set \(S\) at \(a\).

1.5 Definition 4.3.5

Suppose \(f\) is a function and \(a\) is a number in the domain of \(f\).

  • We say that \(f\) has a local minimum at \(a\) if there is some number \(d > 0\) such that the interval \((a-d, a+d)\) is contained in the domain of \(f\), and \(f\) attains its minimum value on \((a-d,a+d)\) at \(a\).
  • We say that \(f\) has a local maximum at \(a\) if there is some number \(d>0\) such that the interval \((a-d,a+d)\) is contained in the domain of \(f\), and \(f\) attains its maximum value on \((a-d,a+d)\) at \(a\).

1.6 Theorem 4.3.6

Suppose \(f\) has a local maximum or minimum at \(a\). Then either \(f\) is not differentiable at \(a\) or \(f'(a) = 0\).

1.7 Definition 4.3.7

A number \(a\) in the domain of \(f\) such that either \(f'(a)\) is undefined or \(f'(a)=0\) is called a critical number of \(f\).

1.8 Theorem 4.3.8 (First Derivative Test)

Suppose that \(f\) is continous at \(a\), and for some number \(d>0\), \(f\) is differentiable on the intervals \((a-d,a)\) and \((a,a+d)\).

  • Suppose that for every number \(x\), if \(a-d < x < a\) then \(f'(x) < 0\), and if \(a < x < a + d\) then \(f'(x) > 0\). Then \(f\) has a local minimum at \(a\).
  • Suppose that for every number \(x\), if \(a-d < x < a\) then \(f'(x) > 0\), and if \(a < x < a + d\) then \(f'(x) < 0\). Then \(f\) has a local maximum at \(a\).

Notes:

  • A function need not have a local maximum or minimu at a critical number.