Mathematics - Application of Derivative Question with Solution | TestHub

Here $\frac{dy}{dx}=3x^2-3$. For statement 1, ${\frac{dy}{dx}|}_{(0,1)}=-3$; the tangent $y=-3x+1$ meets $y=x^3-3x+1$ where $-3x+1=x^3-3x+1\Rightarrow x^3=0\Rightarrow x=0$ (a triple root), so the tangent meets the curve at one point only; statement 1 is true. For statement 2, ${\frac{dy}{dx}|}_{(1,-1)}=0$; the tangent is $y=-1$ and $-1=x^3-3x+1\Rightarrow x^3-3x+2=0\Rightarrow (x-1{)}^2(x+2)=0$, so the tangent meets the curve at two points ( $x=1$ double and $x=-2$ ); statement 2 is false.