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If possible, find constants a and b so that the function f that follows is continuous and differentiable everywhere. If it is not possible, explain why not.

f(x)=ax-b,ifx<2bx2+1,ifxโ‰ฅ2

Short Answer

Expert verified

Ans:b=15&a=45

Step by step solution

01

Step 1. Given information is:

f(x)=ax-b,ifx<2bx2+1,ifxโ‰ฅ2

02

Step 2. Calculating a and b

Differentiatingthefunction,f'(x)=a,ifx<22bx,ifxโ‰ฅ2Sinceitiscontinuousanddifferentiablethereforelefthandderivativeatx=2,isaandrighthandderivativeatx=2isf'(2)=4b.Therefore,a=4b.....(1)Nowthefunctioniscontinuousatx=2Therefore,limxโ†’2-ax+b=limxโ†’2+bx2+12a+b=4b+12(4b)+b=4b+1[From(1)]5b=1b=15Putvalueofbin(1)a=45

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