Question

The number of real values of \[k\] for which the equation \[{x^2} - 3x + k = 0\] has two distinct roots lying in the interval \[\left( {0,1} \right)\] are
A. Three
B. Two
C. Infinitely Many
D. No values of \[k\] satisfies the requirement

Answer 1

Hint: First, we will differentiate the given equation and substitute it equal to 0 to find the value of \[x\]. Then substitute the obtained value of \[x\] in the first-order derivative of the given function \[f\left( x \right)\]. Then we will check if the obtained roots belong in the given interval for the value of \[k\].

Complete step-by-step answer:

Given that the equation is \[f\left( x \right) = {x^2} - 3x + k\].

Differentiating the above function \[f\left( x \right)\] w.r.t \[x\], we get

\[f'\left( x \right) = 2x - 3\]

Now taking \[f'\left( x \right) = 0\], we get

\[
   \Rightarrow 2x - 3 = 0 \\
   \Rightarrow 2x = 3 \\
   \Rightarrow x = \dfrac{3}{2} \\
\]

Substituting this value of \[x\] in the equation \[f'\left( x \right) = 2x - 3\], we get

\[
   \Rightarrow f'\left( {\dfrac{3}{2}} \right) = 2\left( {\dfrac{3}{2}} \right) - 3 \\
   \Rightarrow f'\left( {\dfrac{3}{2}} \right) = 2 - 3 \\
   \Rightarrow f'\left( {\dfrac{3}{2}} \right) = - 1 \\
\]

Thus, we have \[f'\left( {\dfrac{3}{2}} \right) < 0\].
.
We know that if \[f'\left( x \right) < 0\] at each point in an interval, then the function \[f\left( x \right)\] is said to be a decreasing function on that interval.

Thus,\[f\left( x \right)\] is a decreasing function.

Since we know that \[f'\left( x \right) = 0\], which gives us only one point at \[x = \dfrac{3}{2}\], where \[\dfrac{3}{2} \notin \left( {0,1} \right)\].

Therefore, we have found that \[f\left( x \right) = 0\] has no root in \[\left( {0,1} \right)\] for any value of \[k\].

Hence, the correct option is D.

Note: In these types of questions, some students take\[f\left( x \right)\] is a decreasing function for \[f'\left( x \right) < 0\], which is wrong.
 In this question, we can also assume the value of two distinct roots \[a\] and \[b\] from the equation in interval \[\left( {0,1} \right)\] such that they can never add up to 2. Then the roots can either be 1 and 1 or 0 and 2 of the quadratic equation, but both cases don’t satisfy the condition in this question. Thus, there is no real value for which the equation has two distinct roots.