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Q.
The curve $y(x)=a x^3+b x^2+c x+5$ touches the $x$-axis at the point $P (-2,0)$ and cuts the $y$-axis at the point $Q$, where $y$ ' is equal to 3 . Then the local maximum value of $y(x)$ is :
$y(x)=a x^3+b x^2+c x+5$ is passing through $(-2,0)$ then $8 a-4 b+2 c=5 \ldots \ldots$ (1)
$y^{\prime}(x)=3 a x^2+2 b x+c$ touches $x$-axis at $(- 2,0)$
$12 a-4 b+c=0 .....$(2)
again, for $x=0, y^{\prime}(x)=3$
$c =3 .....$(3)
Solving eq. (1), (2) \& (3) $a =-\frac{1}{2}, b =-\frac{3}{4}$
$y^{\prime}(x)=-\frac{3}{2} x^2-\frac{3}{2} x+3$
$y(x)$ has local maxima at $x=1$
$y(1)=\frac{27}{4}$