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

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(B) Let $f(t) = t^2 + 5mt - 3m + 1$,where $t = \{x\} \in [0, 1)$.We require $f(t) < 0$ for all $t \in [0, 1)$.Since $f(t)$ is a quadratic in $t$ with

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$0$

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(B) Let $f(t) = t^2 + 5mt - 3m + 1$,where $t = \{x\} \in [0, 1)$.We require $f(t) Since $f(t)$ is a quadratic in $t$ with a positive leading coefficient,the parabola opens upward.For $f(t) $f(0) = -3m + 1  1 \implies m > \frac{1}{3}$.$f(1) = 1 + 5m - 3m + 1 = 2m + 2 \leq 0 \implies 2m \leq -2 \implies m \leq -1$.Combining $m > \frac{1}{3}$ and $m \leq -1$,we find no such $m$ exists.Thus,the number of integral values of $m$ is $0$.

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

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