Inverse Trigonometric Functions

Step 1: Express

f(x)

in simpler formGiven

f(x)%20%3D%20%5Ccos(%5Ctan%5E%7B-1%7D%20x)

Let

%5Ctheta%20%3D%20%5Ctan%5E%7B-1%7D%20x.

Then, by definition,

%5Ctan%20%5Ctheta%20%3D%20x.

We can now use the identity:

%5Ccos%20%5Ctheta%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B1%2B%5Ctan%5E2%20%5Ctheta%7D%7D

Therefore,

f(x)%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7D

Step 2: Differentiate

f(x)

f'(x)%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft(%20(1%2Bx%5E2)%5E%7B-1%2F2%7D%20%5Cright)%20%3D%20-%5Cfrac%7B1%7D%7B2%7D(1%2Bx%5E2)%5E%7B-3%2F2%7D%20%5Ccdot%20(2x)%20%3D%20-%5Cfrac%7Bx%7D%7B(1%2Bx%5E2)%5E%7B3%2F2%7D%7D

Step 3: Determine where

f'(x)

is negative or positiveSince the denominator

(1%2Bx%5E2)%5E%7B3%2F2%7D%20%3E%200

for all

x

,the sign of

f'(x)

depends only on the numerator

-x

:For

x%20%3E%200

f'(x)%20%3D%20-%5Cfrac%7Bx%7D%7B(1%2Bx%5E2)%5E%7B3%2F2%7D%7D%20%3C%200

⇒

f

isdecreasingfor

x%20%3E%200.

For

x%20%3C%200

f'(x)%20%3E%200

⇒

f

isincreasingfor

x%20%3C%200.

x%20%3D%200

, derivative is 0.Step 4: Choose the correct optionOnly the statement that

f

is decreasing for

x%20%3E%200

is true.✅Correct answer: Option AFinal Answer:

%5Cboxed%7B%5Ctext%7BOption%20A%3A%20%7D%20f%20%5Ctext%7B%20is%20decreasing%20for%20%7D%20x%20%3E%200.%7D

Chapter Questions
1 MCQs