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find the value of a and b if
f(x)= x^2+3x+a, x>=1
bx+2 for, x>1
is differentiable at x=1​

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Find 'a' and 'b'. f(x) = { x2 + 3x + a , x< 1 and bx + 2 , x > 1

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student-name M . S . Rawat answered this

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Find the values of a and b iff(x) = {x2+3x+a, x≤1bx+2, x > 1is differentiable at each x∈R.Here is the solution.RHD at x = 1 = limh→0f(1+h)−f(1)h⇒Rf'(1) =limh→0 b(1+h)−(1+3+a)h⇒Rf'(1) =limh→0b+bh−4−ah⇒Rf'(1) =limh→0b = bLHD at x = 1 = limh→0f(1−h)−f(1)−h⇒Lf'(1) =limh→0(1−h)2+3(1−h)+a−4−a−h⇒Lf'(1) =limh→0h2−5h−h =limh→0(5−h) = 5Since f is differentiable at x = 1, so Rf'(1) = Lf'(1)⇒b = 5Since every differentiable function is continuous, so f is continuous at x = 1.Now, LHL = limx→0−f(x) = limx→0−(x2+3x+a)put x = 1−h, as x→1−, h→0LHL = limh→0(1−h)2+3(1−h)+a = limh→0(h2−5h+4+a) = 4+aRHL = limx→0+f(x) = limx→0+ bx+2put x = 1+h, as x→1+, h→0RHL = limh→0[b(1+h)+2] = b+2Now, f(1) = 4+aSince f is continuous at x = 1, thenLHL = RHL = f(1)⇒LHL = RHL⇒4+a = b+2⇒4+a = 5+2⇒a = 3

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