idku
  • idku
just reviewing something from calc 1 (made up ex.)
Mathematics
schrodinger
  • schrodinger
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idku
  • idku
Should I make a function up? \(\tt f(x)=x^3+3x^2+2x+1\) now will do a lot of things to it...
idku
  • idku
At first, I will just simply determine when the function is increasing and when decreasing, using the first derivative.
idku
  • idku
\(\tt f'(x)=3x^2+6x+2\) This is the derivative, or the slope, of the function \(\tt f(x)\). We know from elementary maths before calculuses, that: \(\bullet\) When the slope is positive the function is increasing \(\bullet\) When the slope is negative the function is decreasing

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idku
  • idku
So, we will set \(\tt f'(x)>0\) to find where the function is increasing, and set \(\tt f'(x)<0\) to find where the function is decreasing.
idku
  • idku
now, won't actually solve that... i need the concepts.
idku
  • idku
Ok, now, the critical numbers.
idku
  • idku
the critical numbers are the values of \(\tt f(x)\), are when: 1. \(\tt f'(x)=0\) (will have to set the derivative =0 and solve for x) 2. \(\tt f'(x)\) is undefined (provided that \(\tt f(x)\) has a value at that point) Polynomial is always continuous, and even without knowing this fact, I can tell that no restrictions in \(\tt f(x)\) or \(\tt f'(x)\) exist.
idku
  • idku
Then \(\tt f'(b)=greatest~output~of~B\) \(\tt f'(a)=smallest~output~of~A\) when i check critical numbers, that would mean that I have absolute maximum of B at x=b absolute minimum of A at x=a

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