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Maths-

General

Easy

Question

On the interval [0, 1] the function $x to the power of 25 end exponent open parentheses 1 minus x close parentheses to the power of 75 end exponent$ takes its maximum value at the point

0
$\frac{1}{4}$
$\frac{1}{2}$
$\frac{1}{3}$

The correct answer is: $fraction numerator 1 over denominator 4 end fraction$

Given, $f open parentheses x close parentheses equals x to the power of 25 end exponent open parentheses 1 minus x close parentheses to the power of 75 end exponent$
$rightwards double arrow blank f to the power of ´ end exponent open parentheses x close parentheses equals 25 x to the power of 24 end exponent open parentheses 1 minus x close parentheses to the power of 75 end exponent minus 75 x to the power of 25 end exponent open parentheses 1 minus x close parentheses to the power of 74 end exponent$

$equals 25 x to the power of 24 end exponent open parentheses 1 minus x close parentheses to the power of 74 end exponent left parenthesis 1 minus 4 x right parenthesis$

Put, $f to the power of ´ end exponent open parentheses x close parentheses equals 0 blank rightwards double arrow blank x equals 0 comma blank 1 comma fraction numerator 1 over denominator 4 end fraction$

If $x less than fraction numerator 1 over denominator 4 end fraction$ , then

$f to the power of ´ end exponent open parentheses x close parentheses equals 25 x to the power of 24 end exponent open parentheses 1 minus x close parentheses to the power of 74 end exponent open parentheses 1 minus 4 x close parentheses greater than 0$

And if $x greater than fraction numerator 1 over denominator 4 end fraction$ , then

$f to the power of ´ end exponent open parentheses x close parentheses equals 25 x to the power of 24 end exponent open parentheses 1 minus x close parentheses to the power of 24 end exponent open parentheses 1 minus 4 x close parentheses less than 0$

Thus, $f to the power of ´ end exponent open parentheses x close parentheses$ changes its sign from positive to negative as $x$ passes through 1/4 from left to right.

Hence, $f left parenthesis x right parenthesis$ attains its maximum at x=1/4

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