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Q.
If the point $(1,3)$ serves as the point of inflection of the curve $y = ax ^3+ bx ^2$ then the value of ' $a$ ' and ' $b$ ' are -
Application of Derivatives
Solution:
$y=a x^3+b x^2 $
$y^{\prime}=3 a x^2+2 b x $
$y^{\prime \prime}=6 a x+2 b$
for point of inflection $y^{\prime \prime}=0$
$x=\frac{-b}{3 a} $
$3 a+b=0$(i) $($ as $x =1)$
point satisfy the curve also, so
$3= a + b$.....(ii)
from (i) & (ii)
$a =-\frac{3}{2}, b =\frac{9}{2}$