Answer to Question #202083 in Calculus for Nauman Zaffar

Question #202083

For what values of a and b is

g(x)={ax+2b, x<=0

x2+3a-b, 0<x<=2

3x-5, x>2}

continuous at every x?


1
Expert's answer
2021-06-03T08:10:56-0400

if g(x) is continuous at every x then

"\\mathop {\\lim }\\limits_{x \\to 0 - 0} g(x) = \\mathop {\\lim }\\limits_{x \\to 0 + 0} g(x) \\Rightarrow a \\cdot 0 + 2b = {0^2} + 3a - b"

"\\mathop {\\lim }\\limits_{x \\to 2 - 0} g(x) = \\mathop {\\lim }\\limits_{x \\to 2 + 0} g(x) \\Rightarrow {2^2} + 3a - b = 3 \\cdot 2 - 5"

We have a system of equations:

"\\left\\{ \\begin{matrix}\n2b = 3a - b\\\\\n4 + 3a - b = 1\n\\end{matrix} \\right."

"a = b = - \\frac{3}{2}"

Answer: "a = b = - \\frac{3}{2}"


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