What is m if function f(x) have primitives? f(x) = 2x+a^2+sinx, ifx< or equal 0 f(x) =xcosx+e^x, if x>0

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The content of the problem is vague since it requests for you to find the parameter m but there exists no such a parameter in any of the two equations of the function f(x). Instead of m, there exists the parameter a, hence supposing that you do need to find the parameter that is present in equation of function, under the condition provided, you need to remember that all continuous and differentiable functions have primitives.

You need to test the continuity of the function at `x = 0` , hence, you need to evaluate the limit of the function at `x = 0` and then you need to evaluate the value of function at `x = 0` . If these values coincide, the function is continuous.

Starting with evaluation of the limit when x goes to 0, yields:

`lim_(x->0,x<0) (2x+a^2+sinx) = 2*0 + a^2 + sin 0`

`lim_(x->0,x<0) (2x+a^2+sinx) = a^2`

`lim_(x->0,x>0) (x*cos x + e^x) = 0*cos 0 + e^0`

`lim_(x->0,x>0) (x*cos x + e^x) = 1`

Evaluating the value of he function at `x = 0` , yields:

`f(0) = 2*0 + a^2 + sin 0 => f(0) = a^2`

Equating `lim_(x->0,x<0) (2x+a^2+sinx) = lim_(x->0,x>0) (x*cos x + e^x) = f(0)` , yields:

`a^2 = 1 => a = +-1`

Hence, evaluating the parameter a, using the continuity condition for the function to have primitives, yields `a = +-1` .