Verify Rolles theorem for each of the following functions on the indicated intervals: `f(x)=x(x+3)e^(-x/2)on[-3,0]` `f(x)=e^x(sinx-cosx)on[pi/4,(5pi)/4]`

Explanation

Given,

`f(x)=e^x(sinx−cosx),x in[pi/4,(5pi)/4]`

Sine, cosine and exponential function are always continuous.

so, Given function is continuous in `[pi/4,(5pi)/4]`

Differentiating by x, we get
`f′(x)=e^x(cosx+sinx)+(sinx−cosx)e^x`

=`e^x[cosx+sinx+sinx−cosx]`

`=2e^xsinx`

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