Question

Investigate the behaviour of the function $y=(x^3+4)(x+1{)}^3$ and construct its graph. How many solutions does the equation $(x^3+4)(x+1{)}^3=c$ possess ?

Answer 1

$y=(x^3+4){(x+1)}^3$ ...... $\left(1\right)$

$\Rightarrow$point of intersection on axis

At $x=0,y=4\times {\left(1\right)}^2=4$

At $y=0,(x^3+4){(x+1)}^3=0$

$x=-1$

$x={(-4)}^{1/3}$

Increasing and decreasing nature

$\frac{dy}{dx}=(x^3+4)\times 3{(x+1)}^2+{(x+1)}^3\left(3x^2\right)$

$\frac{dy}{dx}={(x+1)}^2[3x^3+12+3x^2]$

$\frac{dy}{dx}=3{(x+1)}^2[x^3+x^2+4]$ ..... $\left(2\right)$

At $x=-1,-2$ from above turning point exist

For maxima and minima

$y=[{(-1)}^3+4]{(-1+1)}^3,y=0$

$y=[{(-2)}^3+4]{(-2+1)}^3,y=4$

(Image)

$(x^3+4){(x+1)}^3=C$

So, $2$ solutions exist.