Number of integral values of m for which {x}^ | vedclass

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(B) Let $f(t) = t^2 + 5mt - 3m + 1$,where $t = \{x\}$. Since $x \in R$,$t$ takes all values in the interval $[0, 1)$.We are given $f(t) < 0$ for all $

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$0$

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(B) Let $f(t) = t^2 + 5mt - 3m + 1$,where $t = \{x\}$. Since $x \in R$,$t$ takes all values in the interval $[0, 1)$.We are given $f(t) Since $f(t)$ is a quadratic in $t$ with a positive leading coefficient,the parabola opens upwards.For $f(t) Evaluating at $t = 0$: $f(0) = -3m + 1  1 \implies m > 1/3$.Evaluating at $t = 1$: $f(1) = 1 + 5m - 3m + 1 = 2m + 2 Since there is no value of $m$ that satisfies both $m > 1/3$ and $m < -1$,there are no such integral values of $m$.

Number of integral values of $m$ for which ${x}^2 + 5m{x} - 3m + 1 < 0$ $\forall x \in R$ is (where ${.}$ denotes the fractional part function).

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