Extrema of f(x) − x² from a Graph
Difficulty: Hard · Topics: Calculus, Derivatives, Graphs
Grafikdan f(x) − x² ning ekstremumlari
The graph of the quartic function y=f(x) is shown below.
graph "x":[-3.9,4.9],"y":[-1.6,5.0],"xTicks":[-2,1,3],"yTicks":[],"label":"Graph of a quartic y = f(x) with local maxima at x = -2 and x = 3 and a local minimum at x = 1","items":["type":"curve","points":[[-3.375,-1.335],[-3.25,-0.218],[-3.125,0.744],[-3.0,1.562],[-2.875,2.248],[-2.75,2.811],[-2.625,3.262],[-2.5,3.611],[-2.375,3.867],[-2.25,4.039],[-2.125,4.136],[-2.0,4.167],[-1.875,4.139],[-1.75,4.06],[-1.625,3.937],[-1.5,3.777],[-1.375,3.587],[-1.25,3.373],[-1.125,3.141],[-1.0,2.896],[-0.875,2.643],[-0.75,2.386],[-0.625,2.131],[-0.5,1.882],[-0.375,1.64],[-0.25,1.411],[-0.125,1.197],[0.0,1.0],[0.125,0.823],[0.25,0.666],[0.375,0.533],[0.5,0.423],[0.625,0.338],[0.75,0.277],[0.875,0.241],[1.0,0.229],[1.125,0.241],[1.25,0.274],[1.375,0.329],[1.5,0.402],[1.625,0.492],[1.75,0.596],[1.875,0.711],[2.0,0.833],[2.125,0.96],[2.25,1.086],[2.375,1.207],[2.5,1.319],[2.625,1.416],[2.75,1.493],[2.875,1.544],[3.0,1.562],[3.125,1.542],[3.25,1.475],[3.375,1.355],[3.5,1.173],[3.625,0.922],[3.75,0.594],[3.875,0.178],[4.0,-0.333],[4.125,-0.95]],"start":"arrow","end":"arrow","type":"guide","at":[-2,4.167],"to":"x","type":"guide","at":[1,0.229],"to":"x","type":"guide","at":[3,1.562],"to":"x","type":"label","at":[-3.0,4.4],"text":"y = f(x)"]
Let g(x)=f(x)-x^2. The quartic g(x) has local extrema at x=α, x=β and x=gamma. Find f'(α)× f'(β)× f'(gamma).
To‘rtinchi darajali y=f(x) funksiyaning grafigi quyida ko‘rsatilgan.
graph "x":[-3.9,4.9],"y":[-1.6,5.0],"xTicks":[-2,1,3],"yTicks":[],"label":"Graph of a quartic y = f(x) with local maxima at x = -2 and x = 3 and a local minimum at x = 1","items":["type":"curve","points":[[-3.375,-1.335],[-3.25,-0.218],[-3.125,0.744],[-3.0,1.562],[-2.875,2.248],[-2.75,2.811],[-2.625,3.262],[-2.5,3.611],[-2.375,3.867],[-2.25,4.039],[-2.125,4.136],[-2.0,4.167],[-1.875,4.139],[-1.75,4.06],[-1.625,3.937],[-1.5,3.777],[-1.375,3.587],[-1.25,3.373],[-1.125,3.141],[-1.0,2.896],[-0.875,2.643],[-0.75,2.386],[-0.625,2.131],[-0.5,1.882],[-0.375,1.64],[-0.25,1.411],[-0.125,1.197],[0.0,1.0],[0.125,0.823],[0.25,0.666],[0.375,0.533],[0.5,0.423],[0.625,0.338],[0.75,0.277],[0.875,0.241],[1.0,0.229],[1.125,0.241],[1.25,0.274],[1.375,0.329],[1.5,0.402],[1.625,0.492],[1.75,0.596],[1.875,0.711],[2.0,0.833],[2.125,0.96],[2.25,1.086],[2.375,1.207],[2.5,1.319],[2.625,1.416],[2.75,1.493],[2.875,1.544],[3.0,1.562],[3.125,1.542],[3.25,1.475],[3.375,1.355],[3.5,1.173],[3.625,0.922],[3.75,0.594],[3.875,0.178],[4.0,-0.333],[4.125,-0.95]],"start":"arrow","end":"arrow","type":"guide","at":[-2,4.167],"to":"x","type":"guide","at":[1,0.229],"to":"x","type":"guide","at":[3,1.562],"to":"x","type":"label","at":[-3.0,4.4],"text":"y = f(x)"]
g(x)=f(x)-x^2 bo‘lsin. To‘rtinchi darajali g(x) funksiya x=α, x=β va x=gamma da lokal ekstremumlarga ega. f'(α)× f'(β)× f'(gamma) ni toping.