The curve y= a x^3+ b x^2+ c x+5 touches the x -axis at (-2,0) and cuts the y -axis at a point Q where its gradient is 3 , then the value of a + b + c is

Step-by-step solution

aligned & y= a x^3+ b x^2+ c x+5 touches X-axis at (-2,0) & 0=-8 a +4 ~b -2 c +5 & 8 a -4 ~b +2 c =5 aligned Also, it cuts Y -axis at a point Q Put x=0 in the equation of curve, we get aligned & y=5 & y= a x^3+b x^2+ c x+5 & ~d y ~d x =3 a x^2+2 ~b x+ c & ( ~d y ~d x )_ Q (0,5) =3 & 3 a (0)^2+2 ~b (0)+ c =3 & c =3 aligned Equation (i) becomes, aligned & 8 a-4 b+6=5 & 8 a-4 b+1=0...(ii) aligned Also, X -axis is tangent to curve aligned & ( d y d x )_ x=-2 =0 & 3 a(-2)^2+2 b(-2)+3=0 & 12 a-4 b+3=0...(iii) aligned Sol