Question

Find the points of maxima, minima and the intervals of monotonicity of the following function:
$y=x^4+4x^3-8x^2+3$

Answer 1

$y=x^4+4x^3-8x^2+3$

For montonicity (Increasing and decreasing function)

$\Rightarrow \frac{dy}{dx}=4x^3+12x^2-16x$

$=4x(x^2+3x-4)>0$

$\Rightarrow x(x^2+4x-3x-4)>0$

$x\left[x\right(x+4)-1(x+4\left)\right]>0$

$x(x-1)(x+4)>0$ so

$x\in (-4,0)\cup (1,\infty )$

and decreasing $x\in (-\infty ,-4)\cup (0,1)$

for maxima and minima

$f^{\prime }\left(x\right)=0$

$4x(x^2+3x-4)=0$

$\Rightarrow 4x(x-1)(x+4)=0$

$\Rightarrow x=0,1,-4$

$f^{\prime \prime }\left(x\right)=12x^2+24x-16$

Now at $x=0$

$f^{\prime \prime }\left(0\right)=-16$ (local maxima)

$x=1$

$f^{\prime \prime }\left(1\right)>0$ (local minima)

$x=4$

$f^{\prime \prime }(-4)>0$ (local minima).