1
$\begingroup$

I am reading through page 278 of the textbook "Calculus: Early Transcendentals" 8th Edition, by James Stewart. I am confused by his use of the word "hypothesis" here. I thought a hypothesis was essentially an educated guess?

enter image description here

What does the author mean by "hypothesis" here?

$\endgroup$
8
  • 4
    $\begingroup$ The assumptions of the theorem. $\endgroup$
    – Minus One-Twelfth
    Mar 21, 2019 at 20:18
  • 2
    $\begingroup$ Hypothesis here means the assumptions of the theorem, the 'if' part. For example, if the theorem goes, "If $f$ is a continuous function on a closed interval, then...", the hypothesis is the assumption of continuity and closed interval. $\endgroup$
    – Shubham Johri
    Mar 21, 2019 at 20:22
  • 2
    $\begingroup$ The theorem assumed certain things (continuity and closed interval). The author is referring to these, saying how the theorem does not hold anymore if one of these assumptions is omitted. $\endgroup$
    – Minus One-Twelfth
    Mar 21, 2019 at 20:22
  • 1
    $\begingroup$ The author wants to say that functions not satisfying the assumptions of the theorem may not have absolute maximum/minimum, whilst the ones that do will necessarily have these extrema. $\endgroup$
    – Shubham Johri
    Mar 21, 2019 at 20:27
  • 1
    $\begingroup$ Any theorem is of the form "If A then B". The "hypothesis" is "A". That has nothing to do with a guess, "educated" or not! $\endgroup$
    – user247327
    Mar 21, 2019 at 20:28

1 Answer 1

Reset to default
2
$\begingroup$

Well you must have read the Extreme Value Theorem. It has "assumption" and "results". The "assumptions" are called hypotheses:

Extreme Value Theorem:

Hypothesis: Let $f$ be a continuous, real-valued function to the closed interval $[a,b]$

Conclusion: Then $f$ attains extreme values on that interval

$\endgroup$
2
  • $\begingroup$ I guess an "extreme value" is a value greater or less than all other values in that interval? $\endgroup$
    – LuminousNutria
    Mar 21, 2019 at 20:31
  • 1
    $\begingroup$ Yes exactly. More rigorously, extreme values would be defined like this for Extreme Value Theorem: there exists $c,d$ in $[a,b]$ such that $f(c)\leq f(x) \leq f(d)$ for all $x$ in $[a,b]$ $\endgroup$
    – NazimJ
    Mar 21, 2019 at 20:37

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .