How many distinct roots of the equation between (1,2)

Question

The following equation has been provided and the question asks for number of distinct roots between $(1,2)$ of the following equation.

$3x^2-12x+11+\frac{1}{5}\left ( x^3-6x^2+11x-6 \right)$

I tried to solve it using the derivative but couldn't proceed further. I took derivative because there is a relationship between number of roots of a function and it's derivative. Please help me with this. Thanks in advance.

Answer 1

Let $$f(x)=3x^2-12x+11+\frac{1}{5}\left ( x^3-6x^2+11x-6 \right).$$ Thus, $$f(x)=\frac{1}{5}(x^3+9x^2-49x+49).$$

It's obvious that $f$ has a negative root.

Also, since $f(1)>0$ and $f(2)<0$, there is a root in $(1,2)$,

but it's an unique root there because there is the last root for $x>2$.

Answer 2

Put $P(x)=x^3-6x^2+11x-6$ It is easy to see that $P(x)=(x-1)(x-2)(x-3)$. Now let $Q(x)$ your polynomial, we have $Q(x)=P^{\prime}(x)+\frac{P(x)}{5}$. Let $f(x)=P(x)\exp(x/5)$. We get that $f^{\prime}(x)=Q(x)\exp(x/5)$. Now $f(1)=f(2)=f(3)=0$. By the mean value theorem, the derivative of $f$ has a zero say $c_2\in ]1,2[$, and another one $c_3$ in $]2,3[$. In addition, we have $f(x)\to 0$ if $x\to -\infty$. Hence $f$ has a local extremum on $]-\infty,1[$, say $c_1$, and the derivative of $f$ is zero at $c_1$. The $c_k$ are zeros of $Q$, they are distincts, and as $Q$ is of degree $3$, we have all zeros of $Q$, and there is only one in $]1,2[$.

Answer 3

Hint:

On taking derivative you know that the function is monotonic in (1,2), So 1 or 0 roots.

Now Integrate the equation and apply Lagrange's Mean Value theorem.