f(x)= x^4 - 4x^3 - 8x^2 + 48x
f'(x)= 4 (x-2)(x+2)(x-3)
f''(x)= 4(3x^2 -6x- 4)
my problem is i am unable to get the concavity.
what would be my intervals?
Hello, Alioyo!
$\displaystyle f(x)\:=\: x^4 - 4x^3 - 8x^2 + 48x$
$\displaystyle f'(x)\:=\: 4x^3-12x^2 - 16x + 48 \:=\: 4 (x-2)(x+2)(x-3)$
$\displaystyle f''(x)\:=\: 4(3x^2 -6x- 4)$
My problem is i am unable to get the concavity. .Why not?
What would be my intervals?
You're taking Calculus and you don't know how to solve a quadratic equation?
You have: .$\displaystyle 3x^2-6x-4 \:=\:0$
Quadratic Formula: .$\displaystyle x \:=\:\frac{6 \pm\sqrt{36+48}}{6} \:=\:\frac{6\pm\sqrt{84}}{6} \:=\:\frac{6\pm2\sqrt{21}}{6} \:=\:\frac{3\pm\sqrt{21}}{3}$
The intervals are: .$\displaystyle \left(\text{-}\infty,\:\tfrac {3 -\sqrt{21}}{3} \right),\;\left(\tfrac{3-\sqrt{21}}{3},\:\tfrac{3+\sqrt{21}}{3}\right),\; \left(\tfrac{3+\sqrt{21}}{3},\:\infty\right)$