JEE Main 2022 (Online) 30th July Evening ShiftQuestion 1
The number of integral values of $b$ for which $2x^3-3x^2-12x+b=0$ has three distinct real roots is ________.
Differentiation and its Application
Answer$26$
Solution
Concept. Write the equation as $g(x)=-b$ with $g=2x^3-3x^2-12x$; the line $y=-b$ meets the graph three times iff it lies strictly between the local minimum and maximum values. Formulas used. (1) $g'(x)=6x^2-6x-12=6(x-2)(x+1)$Step 1 - Extreme values. By (1), the local max is $g(-1)=-2-3+12=7$ and the local min is $g(2)=16-12-24=-20$. Step 2 - Condition.(The figure shows $y$ divided by $5$.) $$-20<-b<7\iff-7<b<20.$$ Step 3 - Count. Integers $b=-6,-5,\dots,19$: $26$ values. Answer: 26