JEE Main 2022 · 25 Jun · Shift 2 · Mathematics Q28

← JEE Main 2022 · 25 Jun · Shift 2

MathematicsInteger+4 / -1Question 28 of 90

Let $f(x)=\left|(x-1)\left(x^2-2x-3\right)\right|+x-3$ , $x\in\mathbb R$ . If $m$ and $M$ are respectively the number of points of local minimum and local maximum of $f$ in the interval $(0,4)$ , then $m+M$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 3

Sign of $(x-1)(x+1)(x-3)$ on $(0,4)$ : positive on $(0,1)$ , negative on $(1,3)$ , positive on $(3,4)$ .

Pieces. On $(0,1)$ and $(3,4)$ : $f=x^3-3x^2-x+3+x-3=x^3-3x^2$ , decreasing on $(0,1)$ and increasing on $(3,4)$ . On $(1,3)$ : $f=-x^3+3x^2+2x-6$ , with $f'=0$ at $x=1+\sqrt{5/3}$ .

Extrema. Minima at $x=1$ and $x=3$ (corners), maximum at $x=1+\sqrt{5/3}$ : $m+M=3$ .

Answer: 3

Topics:Maxima and minima Inequalities and modulus

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JEE Main 2022 · 25 Jun · Shift 2, Mathematics question 28