Let f be a function defined by f(x)= ?(ax^(2)+b) , x<1 and f(x)= |2-x|, x?1 (a)

Q: Let f be a function defined by f(x)= ?(ax^(2)+b) , x

Lim x->1- (ax^2 +b) = a+b lim x->1+ (2-x) = 1 f(x) is continuous so Limx->1- (ax^2 +b)=limx->1+(2-x) a+b = 1 f(x) = |2-x| f(x) is differentiable at x=1 f'(x) = 1 f(x) = ax^2 + b f'(x) = 2ax at x=1 f'(1) = 2a equate 2a =1 a =1/2 b =1-a =1/2 a=b=1/2

ID: 3189075 • Letter: L

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Let f be a function defined by f(x)= ?(ax^(2)+b) , x<1 and f(x)= |2-x|, x?1 (a) Find a and b so that f is both continuous and differentiable at x=1 Use correct limit notation where it is appropriate to do so (b) if a and b are equal to the numbers tou found in part (a), is f(x) differentiable for all real numbers? Justify your answer using correct limit notation. SHOW ALL WORK PLEASE!!!!

Explanation / Answer

Lim x->1- (ax^2 +b) = a+b lim x->1+ (2-x) = 1 f(x) is continuous so Limx->1- (ax^2 +b)=limx->1+(2-x) a+b = 1 f(x) = |2-x| f(x) is differentiable at x=1 f'(x) = 1 f(x) = ax^2 + b f'(x) = 2ax at x=1 f'(1) = 2a equate 2a =1 a =1/2 b =1-a =1/2 a=b=1/2

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Let f be a function defined by f(x)= ?(ax^(2)+b) , x<1 and f(x)= |2-x|, x?1 (a)

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