Number of roots of a function . Let
$\displaystyle g(x)={\mathrm d^{50}\over \mathrm dx^{50}}\left(x^2-1\right)^{50}$
find number of roots of   $g(x)=0$ in $[-10,10]$
I tried to find number of roots using degree of the polynomial but I cannot verify whether they are in $[-10,10]$
 A: As Achille Hui pointed out, we can apply Rolle’s theorem.
Property:
If a polynomial $P(x)$ has $\,n\,$ real roots (counting them with their algebraic multiplicities) on the interval $\,[a,b]\,,\,$ then the polynomial $P’(x)$ has at least $\,n-1\,$ real roots (counting them with their algebraic multiplicities) on the same interval $\,[a,b]\,.$
Proof:
Let $\,x_1,\,x_2,\,\ldots,\,x_r\in[a,b]\,$ be the distinct real roots of the polynomial $\,P(x)\,$ and let $\,n_1,\,n_2,\,\ldots,\,n_r\,$ be their algebraic molteplicities.
By hypothesis, it results that $\,n_1+n_2+\ldots+n_r=n\,.$
The polynomial $\,P’(x)\,$ has the following distinct real roots $\,x_1,\,x_2,\,\ldots,\,x_r\in[a,b]\,$ respectively with algebraic multiplicities $\,n_1-1,\,n_2-1,\,\ldots,\,n_r-1\,.$
Moreover, for any $\,i\in\big\{1,2,\ldots,r-1\big\}\,,\,$ by applying Rolle’s theorem to the function $\,f(x)=P(x)\,$ on the interval $\,[x_i,x_{i+1}]\,,\,$ we get that there exists $\,c_i\!\in\,]x_i,x_{i+1}[\,\subset[a,b]\,$ such that $\,P’(c_i)=0\,.$
Hence, the polynomial $\,P’(x)\,$ has at least
$(n_1-1)+(n_2-1)+\ldots+(n_r-1)+r-1=\\=n-r+r-1=n-1\,$
real roots (counting them with their algebraic multiplicities) on the interval $\,[a,b]\,$.
Comment to the property:
We cannot prove that $\,P’(x)\,$ has exactly $\,n-1\,$ real roots (counting them with their algebraic multiplicities) on the interval $\,[a,b]\,$ because there are counterexamples.
$P(x)=\dfrac15x^5-2x^4+\dfrac{23}3x^3-14x^2+12x-\dfrac{18}5$
has two complex conjugate roots and $\,n=3\,$ real roots on the interval $\,[0,3]\,,\,$ that are, $\,x_1\!\in\,]0,1[\,$ and $\,x_2=3\,$ respectively with algebraic multiplicities $\,n_1=1\,$ and $\,n_2=2\,.$
Moreover,
$\begin{align}P’(x)&=x^4-8x^3+23x^2-28x+12=\\&=(x-1)(x-2)^2(x-3).\end{align}$
has $\,4>n-1=2\,$ real roots (counting them with their algebraic multiplicities) on the interval $\,[0,3]\,.$

Let $\;\displaystyle g(x)={\mathrm d^{50}\over \mathrm dx^{50}}\left(x^2-1\right)^{50}\;$ and let $\;P(x)=\left(x^2-1\right)^{50}\,.$
$P(x)\,$ has $\,n=100\,$ real roots (counting them with their algebraic multiplicities) on the interval $\,[-1,1]\,.$
By applying the Property for $\,50\,$ times, we get that $\,g(x)=P^{(50)}(x)\,$ has at least $\,n-50=50\,$ real roots (counting them with their algebraic multiplicities) on the interval $[-\!1,\!1].$
Since the degree of the polynomial $\,g(x)\,$ is $\,50\,,\,$ it follows that all the $\,50\,$ roots of $\,g(x)\,$ are real numbers and they all belong to the interval $\,[-1,1]\,.$
Consequently, the number of roots of $\,g(x)\,$ on the interval $\,[-1,1]\,$ is $\,50\,.$
A: You are on the right track. It turns out that all $50$ roots are in $[-10,10]$. You can see this by trying small examples such as the second or third derivative.
