If are roots of the equation with

MATHEMATICS · Single correct · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

If x1,x2,x3x_1,x_2,x_3 are roots of the equation x3−3x+1=0x^3-3x+1=0 with x1<x2<x3x_1<x_2<x_3, then which of the following is/are correct? [Note: [k][k] denotes greatest integer less than or equal to kk ] (a) All roots are real and distinct, (b) x1∈(−2,−1), x2∈(0,12), x3∈(1,2)x_1\in(-2,-1),\,x_2\in(0,\frac{1}{2}),\,x_3\in(1,2) (c) Distance between the local maximum and local minimum of the curve y=x3−3x+1y=x^3-3x+1 is 2. (d) [x1]+[x2]+[x3]=−1[x_1]+[x_2]+[x_3]=-1

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