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Suppose $f$ is differentiable on an interval $I$ and $f^{\prime}(x)>0$ for all numbers $x$ in $I$ except for a single number $c .$ Prove that $f$ is increasing on the entire interval $I.$

Decreasing behavior in the given interval

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

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04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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Suppose $f$ is differentia…

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Suppose $ f $ is different…

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01:57

Show that $f$ is different…

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Let $f$ be differentiable …

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Use one of the theorems in…

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Prove that if $f(x)$ is di…

in this question asked us to prove that it is increasing on the entire trouble and the we already know if prime is positive is strictly positive. Over the interval I except for a single point se xy close today. So, um so if has to be increased, has to be increasing function Montagne IQ increase increasing function over the whole interval because that if it's not so, we proved this by contradiction. If it's not, then name used f is Deke waiting at somewhere. So if f so f is he crazy? It's, um, Interval Small interval Safer own, uh, from X one to x two. So if F is decreasing them is the frosting maturity of off. If it's negative on this intervals moment of our X one x two and then that means we have you forgiven many points we can We can find you infinitely many points. Oh, such that this first of the year a cave off excess negative which contracted to our assumption because I was assumptions that if prime it's great and zero except for a single point now if every sti creating somewhere, that means we can find you think be many points such that the first of the narratives that zero. So it's force, Um, so we can crew that if is increasing. Oh, the whole in terrible I

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