Question

Let the function
$f\left(\theta \right)=\left|\begin{array}{ccc}sin\theta & cos\theta & tan\theta \\ sin\left(\frac{\pi }{6}\right)& cos\left(\frac{\pi }{6}\right)& tan\left(\frac{\pi }{6}\right)\\ sin\left(\frac{\pi }{3}\right)& cos\left(\frac{\pi }{3}\right)& tan\left(\frac{\pi }{3}\right)\end{array}\right|$
where $\theta ϵ[\frac{\pi }{6},\frac{\pi }{3}]$ and $f^{\prime }(\theta$ denote the derivative of $f$ with respect to $\theta$. Which of hte following statement is/are TRUE?
I. There exists $\theta ϵ[\frac{\pi }{6},\frac{\pi }{3}]$ such that $f^{\prime }(\theta =0$
II. There exists $\theta ϵ[\frac{\pi }{6},\frac{\pi }{3}]$ such $f^{\prime }(\theta \ne =0$

• A. I only
• B. II only
• C. Both I and II
• D. Neither I nor II

Answer 1

The correct option is C Both I and II

$f\left(\theta \right)=\left|\begin{array}{ccc}sin\theta & cos\theta & tan\theta \\ sin\left(\frac{\pi }{6}\right)& cos\left(\frac{\pi }{6}\right)& tan\left(\frac{\pi }{6}\right)\\ sin\left(\frac{\pi }{3}\right)& cos\left(\frac{\pi }{3}\right)& tan\left(\frac{\pi }{3}\right)\end{array}\right|$
fuction $f\left(\theta \right)$ is continuous in $[\frac{\pi }{6},\frac{\pi }{3}]$ and is differentiable in $(\frac{\pi }{6},\frac{\pi }{3})$ as it is composite trignometirc function
Also, $\left(\frac{\pi }{6}\right)=f\left(\frac{\pi }{3}\right)=0$
So, By Rolle's theroem, there exists at least one value $\theta ϵ[\frac{\pi }{6},\frac{\pi }{3}]$ such that $f^{\prime }\left(\theta \right)=0$ i.e., (I) is true
Now, $f^{\prime }\left(\theta \right)\ne 0\therefore$ tangent at, at least one point must be other than horizontal, which is obviously true as given funciton $f\left(\theta \right)$ is not a constant function.