Fermat's Theorem - Application of Derivatives - Calculus 1

let's talk about firat theorem fats theorem states that if F ofx has a local maximum or minimum at C and if fime of C exists then frime of C is equal to zero so let's focus on the graph on the left at this point we have a local maximum so the first part checks out now

let's look at the second part at Point C does frpr of C exists we know the function is continuous at Point C there's no jumps or breaks in the graph but is it differentiable if the graph has a smooth turn as opposed to a sharp turn it's going to be differentiable meaning you could take

the the derivative of this function and get a value for it so the fact that it's differentiable at Point C that means that fime of C exists at this point now according to Fat's theorem if these conditions are met if F ofx has a local maximum at Point C and if it's differentiable that means if fim of C

exists at that point then the conclusion is that the derivative at that point will be zero in other words the slope of the tangent line frpr of C will be equal to zero and the slope of any horizontal line will always be zero so we could see the validity of that statement now let's look at the graph on

the right side so at Point C we have a local minimum so the first part checks out now the next part does frime of C exists at Point C what would you say we know it's continuous at Point C there's no jumps or points of discontinuity there but is it differentiable at Point

C the answer is no it's not differentiable at Point C we don't have a smooth turn at this point we have a sharp turn here the slope is negative because the function is decreasing and it's becoming more negative then instantaneously the slope changes to a positive value so imagine let's put some numbers