We have #f:RR->RR,f(x)=x^2-(m-1)x+3m-4;m inRR.#Which are values of #m# for which #f(x)<0,forall x in(0,1)#?

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Answer 1 · 2017-12-30T09:48:40

#m <= 1#

Given:

#f(x) = x^2-(m-1)x+3m-4#

Note that due to this being an upright parabola, #f(x) < 0, AA x in (0, 1)# if and only if both:

#f(0) <= 0" "# and #" "f(1) <= 0#

Evaluating #f(0)# and #f(1)#, these conditions become:

#3m-4 <= 0" "# and hence #m <= 4/3#

#2m-2 <= 0" "# and hence #m <= 1#

Both of these conditions hold if and only if #m <= 1#

graph{x^2-(1-1)x+3(1)-4 [-2.427, 2.573, -1.3, 1.2]}