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Calculus › Extrema and curve sketching › Bounds from derivatives

Maximum integer k in a derivative inequality

High school · ●●●●○ 100 pts

Let f(x)=ex−x−2f(x)=e^{x}-x-2. For x>0x>0, the inequality (x−k)f′(x)+x+1>0(x-k)f'(x)+x+1>0 always holds, where kk is an integer. Find the maximum value of kk.

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derivativesexponential functionsinequalities with parametersmonotonicityintermediate value theorem