7 votes 7 votes Let $I=(a, b)$ be an open interval and let $f$ be a function which is differentiable on $I$. Which of the followings is/are true statements - If $f^{\prime}(x)=0$ for all $x \in I$, then there is a constant $r$ such that $f(x)=r$ for all $x \in I$. If $f^{\prime}(x)>0$ for all $x \in I$, then $f(x)$ is strictly increasing on $I$. If $f^{\prime}(x)<0$ for all $x \in I$, then $f(x)$ is strictly decreasing on $I$. If $f^{\prime}(x)>0$ for all $x \in I$, then $f(x)$ is strictly decreasing on $I$. Calculus goclasses2024-calculus-1 goclasses calculus differentiation maxima-minima multiple-selects 2-marks + – GO Classes asked Aug 28, 2022 • edited Apr 29, 2023 by Lakshman Bhaiya GO Classes 573 views answer comment Share Follow See all 3 Comments See all 3 3 Comments reply PARTH PRAJAPATI commented Oct 6, 2023 reply Follow Share Why option A is correct? 0 votes 0 votes akashrawat commented Oct 19, 2023 reply Follow Share We can’t say about continuity at $a$ and $b$, but only in interval $I=(a,b)$.Since it is asked for $(a,b)$ , we can conclude the value will be constant for all $x\in (a,b)$ as $f’(x) =0$ for all $x\in (a,b)$.However if it was asked for $[a,b]$ , then we can’t say anything, as at $a$ or $b$ the function may be discontinuous and may have a value other than $r$ or not have a value at all. 2 votes 2 votes PARTH PRAJAPATI commented Oct 19, 2023 reply Follow Share Thank you !!🙏 0 votes 0 votes Please log in or register to add a comment.
1 votes 1 votes option B is called local minima and option C is called local maxima Priyotosh2001 answered Jul 28, 2023 Priyotosh2001 comment Share Follow See all 0 reply Please log in or register to add a comment.