$\"\"$

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The function $\"\"$ be a twice-differentiable function and one to one on open interval $\"\"$ .

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Inverse function of $\"\"$ is $\"\"$ .

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From the definition of inverse function,

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$\"\"$ , where $\"\"$ .

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Consider $\"\"$ .

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$\"\"$ .

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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Chain rule: $\"\"$ .

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$\"\"$

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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Substitute $\"\"$ in the above expression.

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$\"\"$

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$\"\"$ .

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$\"\"$

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If $\"\"$ is increasing and concave downward then $\"\"$ and $\"\"$ .

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Hence if $\"\"$ , then $\"\"$ .

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Therefore, the inverse function $\"\"$ is concave upward.

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$\"\"$

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$\"\"$ .

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The inverse function $\"\"$ is concave upward.