A curve with equation of the for y=ax4+bx3+c+cx+d has zero gradient at the point (0, 1) also touches the x-axis at the point (- 1, 0) then the values of then x for which the curve has a negative gradient are:

We have,

$\frac{d y}{d x} = 0$ at $(0, 1)$ and $(- 1, 0)$

$∴ c = 0 and - 4 a + 3 b - c = 0 \Rightarrow a = \frac{3 b}{4} and c = 0$

Also, the curve passes through $(0, 1)$ and $(- 1, 0)$

$∴ d = 1 and 0 = a - b - c + d \Rightarrow a - b - c + 1 = 0$

From (i) and (ii), we get

$\begin{aligned} a = 3, b = 4, c = 0 and d = 1 \\ ∴ y = 3 x^{4} + 4 x^{3} + 1 \Rightarrow d y = 12 x^{3} + 12 x^{2} \end{aligned}$

Now,

$\frac{d y}{d x} < 0 \Rightarrow 12 x^{3} + 12 x^{2} < 0 \Rightarrow x + 1 < 0 \Rightarrow x < - 1$

A curve with equation of the for $y = a x^{4} + b x^{3} + c + c x + d$ has zero gradient at the point $(0, 1)$ also touches the x-axis at the point $(- 1, 0)$ then the values of then x for which the curve has a negative gradient are: