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Let the function

$$f(\theta) = \begin{vmatrix} \sin\theta & \cos\theta & \tan\theta  \\ \sin(\frac{\pi}{6}) & \cos(\frac{\pi}{6}) & \tan(\frac{\pi}{6}) & \\ \sin(\frac{\pi}{3}) & \cos(\frac{\pi}{3}) & \tan(\frac{\pi}{3})   \end{vmatrix} $$

where 

$\theta \in \left[ \frac{\pi}{6},\frac{\pi}{3} \right]$ and $f'(\theta)$     denote the derivative of $f$ with respect to $\theta$. Which of the following statements is/are TRUE?

  1. There exists $\theta \in (\frac{\pi}{6},\frac{\pi}{3})$ such that $f'(\theta) = 0$
  2. There exists $\theta \in (\frac{\pi}{6},\frac{\pi}{3})$ such that $f'(\theta)\neq  0$
  1. I only
  2. II only
  3. Both I and II
  4. Neither I Nor II
asked in Calculus by Veteran (79.1k points)  
edited by | 1.2k views
Someone explain it please.. I m getting both (i) and (ii).. since if we take derivative f(x) then the 2nd and the 3rd row is will be 0.. So determinant is 0 which is independent of theta..!!

2 Answers

+14 votes
Best answer

We need to solve this by rolle's theorem, to apply rolle's theorem following 3 conditions should be satisfied:
1) f(x) should be continuous in interval [a, b], 
2) f(x) should be differentiable in interval (a, b), and
3) f(a) = f(b)

If these 3 conditions are satisfied simultaneously then, there exists at least one 'x' such that
f '(x) = 0

So, for the above question, it satisfies all the three conditions, so we can apply rolle's theorem, i.e, there exists 'at least one' theta that gives f '(theta) = 0

Also, the given function is also not a constant function, i.e f '(theta) ≠ 0

So, answer is C

 

answered by Junior (791 points)  
selected by
how can you say that f(x) is continous and differentiable??

secondly,how did you prove the second stateentt as true i.e f'(theta) !=0

plss explain
"In that given interval" it is continuous as well as differentiable. You should read the definitions of continuity and differentiabilty and check whether it satisfies those properties in that interval.

Also, above function is not constant, it depends on theta, there must be definitely "some theta" where it is not equal to zero. If it would have been constant, then it's derivative i.e. F'(theta) will always be zero.
i am not able to prove how it is continue and differentiable..can u pls help??not good in continuity and differentiabilty
some pls explain how it is continuous and differentiable.. here
@ Akriti,

Trignometric functions are differentiable in their domain also if function is differentiable then it is continuos for sure.

Hope it helps.
you mean sinx ,cos x,tanx will be differantiable at any value??
For this question in this range it is continious and differentiable too
@Akriti simple wy to check them is by vissualizing there graph in the specific domain.
+1 vote

Here is the graph of f'(x).

answered by Junior (977 points)  

Its f(X)= 1.21 *sinX - 0.366 cosX + 0.5 tanX (If f(X) is determinant as shown in ques. )

 

can we put directly f(a)=pi/6 and f(b)=pi/3 in given function

by solving this f(a)=f(b) (which is zero)

by this it is proved that given function is continuous as well as differentiable (Rolle's Theorem)

but why ii point is also true ?


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