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Maths-

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Easy

Question

If a differentiable function $f left parenthesis x right parenthesis$ has a relative minimum at $x equals 0$ , then the function $ϕ open parentheses x close parentheses equals f open parentheses x close parentheses plus a x plus b$ has a relative minimum at $x equals 0$ for

All $a$ and all $b$
All $b$ if $a = 0$
All $b > 0$
All $a > 0$

The correct answer is: All $b$ if $a equals 0$

$ϕ to the power of ´ end exponent open parentheses x close parentheses equals f to the power of ´ end exponent open parentheses x close parentheses plus a$
$because blank ϕ to the power of ´ end exponent open parentheses 0 close parentheses equals 0$

$rightwards double arrow blank f to the power of ´ end exponent open parentheses 0 close parentheses plus a equals 0$

$rightwards double arrow blank a equals 0 blank left parenthesis because f to the power of ´ end exponent open parentheses 0 close parentheses equals 0 right parenthesis$

Also, $ϕ to the power of ´ end exponent open parentheses 0 close parentheses greater than 0 blank left parenthesis because f to the power of ´ ´ end exponent open parentheses 0 close parentheses greater than 0 right parenthesis$

$rightwards double arrow blank ϕ to the power of ´ end exponent left parenthesis x right parenthesis$ has relative minimum at $x equals 0$ for all $b$ if $a equals 0$

Related Questions to study

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