Question

What is the minimum value of function
f(x)=tanx+cotx

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Answer 1

Answer:

the minimum value of this function f(x) = tanx + cotx is 2.

Answer 2

Minimum Value of f(x)=tanx+cotx = 2

GIVEN: f(x)=tanx+cotx
TO FIND: Minimum Value.
SOLUTION:

As we know,
To find the minima or maxima value of f(x)

We need to put f(x) to be equal to 0.

Implying that,

f(x) = sec²x + (-cosec²x)

f(x) = 0 implies secx = ± cosecx

That means,

sinx=±cosx                     -------eq1

Also, we know,

x= Nπ ± π/4, N ∈ Z

Now,

We consider two values x= 45° and 135° each satisfying one of the equations in correspondence to eq1.

Therefore,

While putting x = 45

f(45°) = 1+1=2

And putting x = 135,

f(135°) = -1-1 =-2

Therefore the minimum value of f(x) is -2 where x is an integer.

Now,

Let tanx = y  that implies cotx = 1/y

Also,

We know that the Arithmetic mean > the geometric mean.

Hence (y + 1/y)/2 is greater than or equal to √(y)*(1/y)

Implying that,

A minimum positive value of y+1/y ie. tanx + cotx is 2.

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