If both the roots of the quadratic equation x | vedclass

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Question

(A) Let $f(x) = x^2 - mx + 4.$ For the roots to be real and distinct and lie in the interval $[1, 5],$ the following conditions must be satisfied:$1$.

Vedclass · Accepted Answer

$(4, 5)$

Vedclass · Answer

(A) Let $f(x) = x^2 - mx + 4.$ For the roots to be real and distinct and lie in the interval $[1, 5],$ the following conditions must be satisfied:$1$. Discriminant $D > 0: (-m)^2 - 4(1)(4) > 0 \Rightarrow m^2 - 16 > 0 \Rightarrow m \in (-\infty, -4) \cup (4, \infty).$$2$. $f(1) > 0: 1^2 - m(1) + 4 > 0 \Rightarrow 5 - m > 0 \Rightarrow m $3$. $f(5) > 0: 5^2 - m(5) + 4 > 0 \Rightarrow 29 - 5m > 0 \Rightarrow m $4$. Vertex location: $1 Taking the intersection of all these conditions: $m \in (4, 5).$ Since $(4, 5)$ is given in option $A,$ it is the correct answer.

If both the roots of the quadratic equation $x^2 - mx + 4 = 0$ are real and distinct and they lie in the interval $[1, 5],$ then $m$ lies in the interval.

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