If tan ⁡ ( x ) &gt; x. Find interval of x.We encountered this question in the class today. My lecturer asked me to define a function f ( x ) = tan ⁡ ( x ) − x. Then we were asked to find its derivative, f′(x), i.e, sec 2 ⁡ ( x ) − 1, and solve the inequality f ′ ( x ) &gt; 0. The answer was ( 0 , π / 2 ).My question is, if f ′ ( x ) &gt; 0, it just proves that f(x) is increasing in the interval, it doesn't prove if f(x) is always positive there. Ok, agreed that since tan ⁡ 0 = 0, tan ⁡ x &gt; x in ( 0 , π / 2 ) since it's increasing. But there is possibility of loosing a solution right? It could be possible that in an interval tan ⁡ x − x is decreasing but tan(x) is still &gt; x

Question

If   tan  ⁡  (  x  )  &amp;gt;  x. Find interval of x.We encountered this question in the class today. My lecturer asked me to define a function   f  (  x  )  =  tan  ⁡  (  x  )  −  x. Then we were asked to find its derivative, f′(x), i.e,       sec    2    ⁡  (  x  )  −  1, and solve the inequality       f    ′    (  x  )  &amp;gt;  0. The answer was   (  0  ,      π        /    2  ).My question is, if       f    ′    (  x  )  &amp;gt;  0, it just proves that f(x) is increasing in the interval, it doesn&#039;t prove if f(x) is always positive there. Ok, agreed that since   tan  ⁡  0  =  0,   tan  ⁡  x  &amp;gt;  x in   (  0  ,      π        /    2  ) since it&#039;s increasing. But there is possibility of loosing a solution right? It could be possible that in an interval   tan  ⁡  x  −  x is decreasing but tan(x) is still   &amp;gt;  x

Jordyn Valdez · Accepted Answer

Step 1Note that the derivative is positive everywhere it is defined, i.e. for all   x  ≠  (  2  k  +  1  )            π      2      .Step 2For a continuous function and       x    1    &amp;gt;      x    0  , if you know   f  (      x    0    )  &amp;gt;  0 and f increasing, then   f  (      x    1    )  &amp;gt;  0. But if   f  (      x    0    )  &amp;gt;  0 and f decreasing, you can&#039;t conclude about the sign of   f  (      x    1    )

Deanna Gregory · Accepted Answer

Step 1      f    ′    (  x  )  &amp;gt;  0 in the interval   (  0  ,      π    2    ), so f is increasing on this interval. Since   f  (  0  )  =  0 and f is continuous on the interval, f must be positive on the interval.Step 2Note that   f  (      π    2    ) is undefined, and then   f  (      π    2    +  ϵ  ) for some small   ϵ  &amp;gt;  0 is negative. So we can say that f is always positive on the interval you&#039;ve specified.

If tan(x)>x. Find interval of x.

Answered question

Answer & Explanation

New Questions in Integral Calculus