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Maths-

General

Easy

Question

Let $f open parentheses x close parentheses equals x to the power of 3 end exponent minus 6 x to the power of 2 end exponent plus 12 x minus 3$ . Then at $x equals 2 comma blank f left parenthesis x right parenthesis$ has

A maximum
A minimum
Both a maximum and a minimum
Neither a maximum nor a minimum

The correct answer is: Neither a maximum nor a minimum

We have,
$f open parentheses x close parentheses equals x to the power of 3 end exponent minus 6 x to the power of 2 end exponent plus 12 x minus 3$

$rightwards double arrow f to the power of ´ end exponent open parentheses x close parentheses equals 3 x to the power of 2 end exponent minus 12 x plus 12 comma blank f to the power of ´ end exponent open parentheses x close parentheses equals 6 x minus 12$ and $f to the power of ´ ´ ´ end exponent open parentheses x close parentheses equals 6$

We observe that $f to the power of ´ end exponent open parentheses 2 close parentheses equals 0 comma blank f to the power of ´ ´ end exponent open parentheses 2 close parentheses equals 0$ but $f ´ ´ ´ left parenthesis 2 right parenthesis not equal to 0$

So, $x equals 2$ is neither a point of local maximum nor a point of local minimum. In fact, it is a point of inflexion

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