Question

B. Determine the values of a and b so that f is continuous.

Please help!

Answer 1

following are the solution to this question:

Explanation:

Please find the complete question in the attached file.

`\lim_(x\to 2)+f(x) = \lim_(x\to 2)+ (ax^2-bx+3) = 4a-2b+3\\\\ \to 4a-2b+3 = 4\\\\ \therefore \ \ 4a-2b = 1\\\\`

The one-sided limits of F(x) at x = 3 must be equivalent for f(x) to be continuous at x = 3.

`\to \lim_(x\to 3)- f(x) = \lim_(x\to 3)- (ax^2-bx+3) = 9a-3b+3\\\\ \to \lim_(x\to 3)+ f(x) = \lim_(x\to 3)+ (2x-a+b) = 6-a+b\\\\`

So,

`\to 9a-3b+3 = 6-a+b\\\\\therefore\\ \to 10a -4b = 3`

`\to 4a - 2b = 1......(a)\\\to 10a - 4b = 3.......(b)\\`

In equation a multiply the by -2 and then add in the equation b:

`\to -8a + 4b = -2\\\to 10a - 4b = 3\\ \to 2a = 1\\\\ \to a = (1)/(2)\\\\ \to \ 4((1)/(2)) - 2b = 1\\\\ \to 2 - 2b = 1\\\\ \to -2b = -1\\\\ \to b = (1)/(2)`

So, the value of
`a \ and \ b= (1)/(2)`