Determine whether or not f(x) is continuous and/or differentiable at the given value of x=3. If not determine any left or right continuity or differentiability. f(x)={x2−3x&lt;3x+2x≥3

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Determine whether or not f(x) is continuous and/or differentiable at the given value of x=3. If not determine any left or right continuity or differentiability. f(x)={x2−3x&amp;lt;3x+2x≥3

Alara Mccarthy · Accepted Answer

To determine whether or not the function is continuous and differentiable at x=3f(x)={x2−3x&amp;lt;3x+2x≥3Let us first check the continuity of the function at x=3.limx→3−f(x)=limh→0f(3−h)=limh→0[(3−h)2−3]=32−3=9−3=6limx→3+f(x)=limh→0f(3+h)=limh→0[3+h+2]=3+0+2=5And f(3)=3+2=5∴limx→3−f(x)≠limx→3+f(x)=f(3)∴ f is not continuous at x=3 but is right continuous at x=3.Thus f can not be differentiable at x=3.However, f′(x)={2xx&amp;lt;31x≥3So f′(3−)=2(3)=6 and f(3+)=1Therefore it is left as wll as right differentiable at x=3.

Jeffrey Jordon · Accepted Answer

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Determine whether or not f(x) is continuous and/or differentiable at the given v

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