Verify Rolles theorem for the function: `f(x)=x(x+3)e^(-x//2)` on `[-3,\ 0]` .

Explanation

We have` f(x)=x(x+3) e^{-x / 2}`


`\therefore f^{\prime}(x)=(2 x+3) e^{-x / 2}+(x^{2}+3 x) e^{-x / 2}(-\frac{1}{2}) `



`=e^{-x / 2}[2 x+3-\frac{1}{2}[x^{2}+3 x]]=-\frac{1}{2}[x^{2}-x-6] e^{-x / 2}`


`f(x) `exist for every value of` x` in the interval `[-3,0]`
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