Critical number & maximum and minimum values

Lessons

critical number: a number$\ c$ in the domain of a function$\ f$ such that:

• Introduction
  How to describe graphs of functions?
  
  
  a)

  
  $\bullet$ local maximum
  $\bullet$ local minimum
  $\bullet$ critical number
  
  
  
  b)

  
  state the:
  $\bullet$ absolute maximum
  $\bullet$ absolute minimum
  
  
  
  c)

  
  on the interval,
  $-1\leq x\leq 12\,$state the:
  $\bullet$ absolute maximum
  $\bullet$ absolute minimum
  
  
  

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• 1.
  Find the critical numbers of the function:
  a)

  
  $f(x)={^3}\sqrt{x^{2}-5x}$
  
  
  b)

  
  $f(x)=x^{\frac{1}{3}}-x^{-\frac{2}{3}}$
  
  

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• 2.
  First Derivative Test: a test to determine whether or not$\ f$ has a local maximum or minimum at a critical number

  First Derivative Test
  local maximum	local minimum	no maximum or minimum	no maximum or minimum

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• 3.
  $f(x)=3x^{5}-15x^{4}+25x^{3}-15x^{2}+5$
  a)

  Find the critical numbers.
  
  
  b)

  On what intervals is $f$ increasing or decreasing?
  
  
  c)

  Find the local maximum and minimum values.
  
  
  d)

  Sketch the graph.
  
  
  e)

  Find the absolute maximum and minimum values.
  
  

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• 4.
  The Closed Interval Method
  To find the absolute maximum and minimum values of a continuous function $f$ on a closed interval [a, b]:
  1.Find the values of $f$ at the critical numbers of $f$ in (a, b).
  2.Find the values of $f$ at the left-endpoint and right-endpoint of the interval
  3.Compare all values from steps 1 and 2: the largest$\$is the absolute maximum value; the smallest$\$is the absolute minimum value.

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• 5.
  Find the absolute maximum and minimum values of the function:
  
  $f(x)=3x^{5}-15x^{4}+25x^3-15x^2+5$

  $-\frac{1}{2}\leq x\leq\frac{1}{2}$

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