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Maths-

General

Easy

Question

If $f left parenthesis x right parenthesis$ is a function given by $f open parentheses x close parentheses equals open vertical bar table row cell sin invisible function application x end cell cell sin invisible function application a end cell cell sin invisible function application b end cell row cell cos invisible function application x end cell cell cos invisible function application a end cell cell cos invisible function application b end cell row cell tan invisible function application x end cell cell tan invisible function application a end cell cell tan invisible function application b end cell end table close vertical bar comma w h e r e blank 0 less than a less than b less than fraction numerator pi over denominator 2 end fraction$
Then the equation $f to the power of ´ end exponent open parentheses x close parentheses equals 0$

Has at least one root in $(a, b)$
Has at most one root in $(a, b)$
Has exactly one root in $(a, b)$
Has no root in $(a, b)$

The correct answer is: Has at least one root in $left parenthesis a comma b right parenthesis$

We know that $sin invisible function application x comma cos invisible function application x$ and $tan invisible function application x$ are continuous on $open square brackets a comma blank b close square brackets$ and differentiable on $open parentheses a comma blank b close parentheses.$ Therefore, $f left parenthesis x right parenthesis$ is continuous on $open square brackets a comma blank b close square brackets$ and differentiable on $left parenthesis a comma blank b right parenthesis$
Also, $f open parentheses a close parentheses equals f open parentheses b close parentheses equals 0$

Therefore, by Rolle’s theorem there exists at least one $c element of left parenthesis a comma blank b right parenthesis$ such that $f to the power of ´ end exponent open parentheses c close parentheses equals 0$

Hence, $f to the power of ´ end exponent open parentheses x close parentheses equals 0$ has at least one root in $left parenthesis a comma blank b right parenthesis$

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