Study of x³+2x²−4x

Text

Given the function $y = x^3 + 2x^2 - 4x$:

• study the sign and the intersections with the axes;
• find relative maxima and minima;
• find the inflection point and the tangent line at the inflection.

Solution

Zeros and sign. $y = x(x^2 + 2x - 4)$; besides $x=0$, $x^2 + 2x - 4 = 0 \Rightarrow x = -1 \pm \sqrt 5$, i.e. $x \approx 1{,}236$ and $x \approx -3{,}236$. The sign alternates at the three zeros (cubic with positive leading coefficient).

Extrema. $f'(x) = 3x^2 + 4x - 4 = (3x - 2)(x + 2) = 0 \Rightarrow x = -2,\ x = \tfrac23$. At $x=-2$: maximum, $f(-2) = -8 + 8 + 8 = 8$. At $x=\tfrac23$: minimum, $f\!\left(\tfrac23\right) = -\tfrac{40}{27} \approx -1{,}481$.

Inflection and tangent. $f''(x) = 6x + 4 = 0 \Rightarrow$ inflection at $x = -\tfrac23$, with $f\!\left(-\tfrac23\right) = \tfrac{88}{27} \approx 3{,}259$. The slope of the tangent at the inflection is $f'\!\left(-\tfrac23\right) = -\tfrac{16}{3} \approx -5{,}333$. $\boxed{\text{max } (-2,\,8);\quad \text{min } \left(\tfrac23,\,-\tfrac{40}{27}\right);\quad \text{inflection at } x = -\tfrac23}$