Let $f$ be a twice$-$differentiable function with $f(0)=4.$ The derivative of $f$ is given by $f^{\text{'}}(x)=sin(x^2-2x+1)$ for $-2x2.$

$($a$)$ Find all values of $x$ in the interval which $f$ has a critical point. Classify each $as$ the location $of$ a relative minimum, a relative maximum, $or$ neither. Justify your answers.

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$(b)$ Use the line tangent $to$ the graph $offatx=0to$ approximate $f(0\mathrm{.}25).$

$(c)On$ the interval $0x0\mathrm{.}25,f^{\text{'}}(x)>0$ and the approximation found $in$ part $(b)an$ overestimate $oran$ underestimate for $f(0\mathrm{.}25)?$ Give a reason for your answer.

$($D$)$ using the MVT explain why the average rate of chnage over the interval $[-2,2]$ cannot equal $1\mathrm{.}25.$