In this page decreasing function we are going to see how to check the given function is decreasing or not. A function f is said to be decreasing on an interval I if f (x₁) ≥ f (x₂) when x₁ < x₂ in I.
How to find whether the given function is decreasing in the given interval.
Step 1: Find the first derivative
Step 2: Apply random values from the given interval. If the simplified value be positive then we can say the given function is decreasing. If it is negative we can say the given function is decreasing.
Example 1:
Check whether y = 4 - 2x² is decreasing on the interval (0,∞)
As per the procedure first let us find the first derivative.
dy/dx = 0 - 2 (2x)
f '(x) = -4 x
Now let us apply random values from the given interval
x = 2 ∈ (0,∞)
f '(2) = -4 (2)
= -8 < 0
x = 6 ∈ (0,∞)
f '(6) = -4 (6)
= -24 < 0
So the given function is decreasing in the interval (0,∞).
Example 2:
Check whether y = sin x is decreasing on the interval (Π/2,Π)
As per the procedure first let us find the first derivative.
dy/dx = Cos x
f '(x) = Cos x
Now let us apply random values from the given interval
x = 2Π/3 ∈ (Π/2,Π) 2Π/3 = 120
f '(2Π/3) = Cos 2Π/3
= Cos (90 + 30)
= - Sin 30
= -1/2 < 0
x = 5Π/6 ∈ (Π/2,Π) 5Π/6 = 150
f '(5Π/6) = Cos 5Π/6
= Cos (90 + 60)
= - sin 60
= -√3/2 < 0
So the given function is decreasing in the interval (Π/2,Π).
Example 2:
Check whether y = 4 - 2 x is decreasing
As per the procedure first let us find the first derivative.
dy/dx = 0 - 2 (1) decreasing function
f '(x) = -2 < 0
So the given function is decreasing for all real numbers.
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Quote on Mathematics
“Mathematics, without this we can do nothing in our life. Each and everything around us is math.
Math is not only solving problems and finding solutions and it is also doing many things in our day to day life. They are:
It subtracts sadness and adds happiness in our life.
It divides sorrow and multiplies forgiveness and love.
Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?
Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”
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