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Maths-

General

Easy

Question

For the curve $y equals x blank e to the power of x end exponent$ , the point

$x = - 1$ is a point of minimum
$x = 0$ is a point of minimum
$x = - 1$ is a point of maximum
$x = 0$ is a point of maximum

The correct answer is: $x equals negative 1$ is a point of minimum

We have,
$y equals x blank e to the power of x end exponent rightwards double arrow fraction numerator d y over denominator d x end fraction equals e to the power of x end exponent plus x blank e to the power of x end exponent$

For maximum or minimum, we must have

$fraction numerator d y over denominator d x end fraction equals 0 rightwards double arrow e to the power of x end exponent open parentheses 1 plus x close parentheses equals 0 rightwards double arrow x equals negative 1$

Now, $fraction numerator d to the power of 2 end exponent y over denominator d x to the power of 2 end exponent end fraction equals 2 e to the power of x end exponent plus x blank e to the power of x end exponent$

$rightwards double arrow open parentheses fraction numerator d to the power of 2 end exponent y over denominator d x to the power of 2 end exponent end fraction close parentheses subscript x equals negative 1 end subscript equals e to the power of negative 1 end exponent open parentheses 2 minus 1 close parentheses greater than 0$

Hence, $x equals negative 1$ is a point of local minimum

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