$\"\"$

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The cubic function is $\"\"$ .

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Consider $\"\"$ .

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The function is having horizontal tangents at points $\"\"$ and $\"\"$ .

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The point $\"\"$ passes through the curve.

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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The point $\"\"$ passes through the curve.

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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If the curve $\"\"$ has horizontal tangent at $\"\"$ , then slope of the tangent line is zero.

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$\"\"$ .

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The curve $\"\"$ has horizontal tangent at $\"\"$ .

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Therefore, $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in above expression then equate $\"\"$ .

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$\"\"$

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$\"\"$

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The curve $\"\"$ has horizontal tangent at $\"\"$ .

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Therefore, $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in above expression then equate $\"\"$ .

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$\"\"$

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$\"\"$

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Solve $\"\"$ and $\"\"$ to get $\"\"$ and $\"\"$ values.

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Subtract the equation $\"\"$ from $\"\"$ .

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$\"\"$

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Add equations (1) and (2).

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$\"\"$

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Substitute $\"\"$ in above expression.

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$\"\"$

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Substitute $\"\"$ in equation $\"\"$ .

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$\"\"$

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Substitute $\"\"$ , $\"\"$ and $\"\"$ in equation $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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

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$\"\"$ .