Question

If $f\left(x\right)=\left|\begin{array}{ccc}sinx& sina& sinb\\ cosx& cosa& cosb\\ tanx& tana& tanb\end{array}\right|,$

where $0<a<b<\frac{\pi }{2}$
then the equation
$f^{\prime }\left(x\right)=0$ has in the interval $(a,b)$

• A. Atleast one root
• B. Atmost one root
• C. No root
• D. exactly one root

Answer 1

The correct option is A

Atleast one root

Here $f\left(a\right)=\left|\begin{array}{ccc}sina& sina& sinb\\ cosa& cosa& cosb\\ tana& tana& tanb\end{array}\right|=0.\text{Also}f\left(b\right)=0.$
Moreover, as sin x, cos x and tan x are continuos and differentiable in (a, b) for 0 < a < b < $\frac{\pi }{2}$, therefore f(x) is also continuos and differentiable in [a, b]. Hence, by Rolle's theorem, there exists atleast one real number c in (a, b) such that f ' (c) = 0.
Hence (a) is the correct answer.