Fermat’s Theorem

==2024-12-12

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Don’t Format your brain after reading this, plzzzzzzz!!

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Theory

Fermat’s theorem regarding derivatives, often referred to as the stationary points theorem, can be formally stated as follows:

Statement of Fermat's Theorem (Stationary Points)

Let $f:(a,b)\to \mathbb{R}$ be a differentiable function. If $x_o\in (a,b)$ is a point where $f$ attains a local extremum (either a local maximum or minimum), then the derivative of $f$ at that point must be zero i.e, $f^{\prime }(x_o)=0$.

This theorem implies that at points of local extrema, the slope of the tangent line to the curve represented by the function is horizontal, indicating no instantaneous rate of change at that point.

The contra-positive of this statement also holds true: if $f^{\prime }(x_o\ne 0)$, then $f$ does not have a local extrema at $x_o$​. This means that if the derivative is non-zero, the function is either increasing or decreasing at that point, and therefore cannot attain a maximum or minimum there. We talk about Intervals of Function Increase or Decrease in that case.

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PTR