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let `f(x)={x^2+ax+b, 0ltxlt2 and 3x+2 , 2lexlt4 and 2ax+5b , 4lexlt8 `. if f(x) is continuous on close interval`(0,8) `.find value of` a and b.` |
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Answer» Here, value of `f(x)` is changing at `2` and `4`. So, `f(x)` to be continuous, `f(2^-) = f(2^+)` `=>2^2+2a+b = 3(2)+2` `=>4+2a+b = 8` `=>2a+b = 4->(1)` Also, `f(x)` to be continuous, `f(4^-) = f(4^+)` `=>3(4) + 2 = 2a(4) +5b` `=>14 = 8a+5b` `=> 8a+5b = 14->(2)` Multiplying (1) with `4` and then subtracting (1) from (2), `8a+4b-8a-5b = 16 - 14` `=> b = -2` Putting `b = -2` in (1), `2a-2 = 4 => a = 3` So, for `a = 3` and `b = -2`, `f(x)` will be continuous. |
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