Let f: R R be a function defined by f(x)= array ll x^3+2 x^2+x+c, & if x b e^x, & if xb array . , where b and c are integers. If f(x) is differentiable x R , then find the value of (b+c) .

Question

Let f: R R be a function defined by f(x)= array ll x^3+2 x^2+x+c, & if x b e^x, & if x>b array . , where b and c are integers. If f(x) is differentiable x R , then find the value of (b+c) .

Quantrex Academy · Accepted Answer

Correct answer: 1

b=0 and c=1 Note that both x^3+2 x^2+x+c and e^x are differentiable in their domain. So make the function differentiable at x=b . Since f is differentiable at x=b . gathered L.H.D. = R.H.D. ( at x=b) 3 b^2+4 b+1=e^b gathered b is an integer, so LHS of the equation will always be an integer. However RHS will be an integer only if b=0 . If b=0 , LHS = RHS, so b=0 is the solution to this equation. Since f is differentiable at x=b , it is implied that it is also continuous at x=b . aligned _ x b⁻ f(x) & =f(b)= _ x b⁺ f

Let f: R R be a function defined by f(x)= array ll x^3+2 x^2+x+c, & if x b e^x, & if x>b array . , where b and c are integers. If f(x) is differentiable x R , then find the value of (b+c) .

Correct answer

Step-by-step solution

More from Continuity and Differentiability