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(a)

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The function is $\"\"$ .

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The function has local minimum is $\"\"$ .

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Since $\"\"$ ,

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$\"\"$ .

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$\"\"$

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The function is $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$ .

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Since $\"\"$ , $\"\"$ .

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$\"\"$

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Solve the equation (1) and (2).

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Multiply equation (1) with $\"\"$ .

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$\"\"$

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Subtract equation (2) from equation (3).

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in equation (1).

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$\"\"$

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$\"\"$ and $\"\"$ .

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(b)

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The function is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in above function.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Then $\"\"$ .

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Therefore the tangent line to the curve at $\"\"$ has a smallest slope.

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(a) $\"\"$ and $\"\"$ .

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(b) The tangent line to the curve at $\"\"$ has a smallest slope.