How to Prove a Curve Has No Turning Points | Rational Functions | Furthermaths P1 | Unit 2

Intro [00:00] In this lecture, we look at one of the most commonly examined question types in Rational Functions: proving whether a curve has turning points or not. Method 1 Double Discriminant [01:15 ] Fix y as a constant, cross-multiply to get a quadratic in x, then use the discriminant to get an expression in y. Taking the discriminant of that expression a second time tells you whether any y-value produces a repeated root and therefore whether a turning point exists. This method is mathematically equivalent to Method 2. Method 2 Differentiate, then Discriminant [05:22] Use the quotient rule to find dy/dx, set the numerator equal to zero, and use the discriminant to determine whether real solutions exist. This is the standard method expected in most mark schemes. #FurtherMaths #ALevelMaths #Cambridge9231 #FP1 #RationalFunctions #Functiona #EdexcelFurtherMaths #EdexcelMaths #Discriminant #PolynomialDivision #Differentiation #ChainRule #QuotientRule #Modulus #Asymptotes #TurningPoints #CurveSketching #ALevel #Mathematics #PureMathematics #Asymptotes #graph #calculus