Prove that if $12a+6b+4c+3d=0$, then $a+bx+cx^2+dx^3=0$ has a real solution in $(0,1)$ Assume that $a,b,c,d$ are real numbers such that $12a+6b+4c+3d=0$.  
Prove that $a+bx+cx^2+dx^3=0$ has a real solution in $(0,1)$.  
Note : I have no idea ! How is the assumption even related to the statement that the question wants us to prove? I don't understand.  
Second Note ( Edited ) : I know that if $x$ is too large, the equation goes to $+\infty$ and when $x$ is so much below $0$, the equation goes to $-\infty$. Then we can apply mean value theorem and say there exists some point such that on that point, the equation becomes zero. It's ok. But how to show that the root is in $(0,1)$ and what is the use of knowing $12a+6b+4c+3d=0$? 
Thanks in advance.
 A: We know Every cubic equation $dx^3+cx^2+bx+a=0$
with real coefficients and $d\not= 0$ has three solutions (some of which may equal each other if they are real, and two of which may be complex non-real numbers) and at least one real solution. So now we put $d=0$ and we get :
$$cx^2+bx+a=0$$ 
and
 $$12a + 6b+ 4c =0 \to 6a+3b+2c = 0 \to b = \frac{6a+2c}{-3}$$
 Now we want to $\Delta_1 \ge0$ ( to has real root).
$\Delta_1 = b^2-4ac = \frac{36a^2+4c^2+24ac}{9} - 4ac = \frac{36a^2+4c^2-12ac}{9}\ge0\iff 9a^2-3ac+c^2\ge0$
Which it is obvious because $\Delta_2 = -27a^2c^2\le0$
Note : When $\Delta \lt 0 $ quadratic polynomial has same sign as coefficient of $x^2$.(In this case 9) and if $\Delta = 0$ value of quadratic polynomial will be zero at one point and in other points has same sign as coefficient of $x^2$.
A: It turns out that $12a+6b+4c+3d=0$ is the condition that makes Rolle's Theorem applicable. Let 
$$ f(x)=\int_0^x(a+bt+ct^2+dt^3)dt=ax+\frac b2x^2+\frac b3x^3+\frac d4x^4.$$
Then clearly $f(x)$ is continuous in $[0,1]$, differentiable in $(0,1)$, $f(0)=0$, and
$$f(1)=a+\frac b2+\frac b3+\frac d4=\frac{1}{12}(12a+6b+4c+3d)=0.$$ By Rolle's Theorem, there is $\xi\in(0,1)$ such that $f'(\xi)=0$, namely $a+bx+cx^2+dx^3=0$ has a root in $(0,1)$.
A: Here are a few things that might help guide you in the right direction:
Let $f(x)$ = $a+bx+cx^2+dx^3$
If $d\neq 0$ :
1) Is $f$ continuous?
2) What is $\lim_{x\to \infty}$$f(x)$? What is $\lim_{x\to -\infty}$$f(x)$?
3) What do we know about the signs of the above two limits, and what can this tell us about the existence of a zero/root?
If $d=0$ , $f$ has degree $\leq2$
How can we show a quadratic (or linear) function has a zero/root?
