The monotony of a function establishes the increasing or decreasing behaviour of the function.
In order to prove that f(x) is an increasing function, we have to do the first derivative test.
If the first derivative of the function is positive, then the function is increasing.
Let's calculate f'(x):
f'(x) = (x^27+x^25+e^(x^3))'
f'(x) = (x^27)' + (x^25)' + (e^(x^3))'
f'(x) = 27x^26 + 25x^24 + e^(x^3)*(x^3)'
f'(x) = 27x^26 + 25x^24 + 2e^(x^3)*(x^2) > 0
Since each term of the expression of f'(x) is positive, the sum of positive terms is also a positive expression. The expression of f'(x) it's obviously>0, so f(x) is an increasing function
f(x) = x^27+x^25+e^(x^3).
We say the function y = f(x) is incresing if f'(x) is > 0 inthe interval.
so we diffentiate f(x) and find whether positive , or positive under any special circumstances.
f'(x) = (x^27+x^25+e^(x^3)'
f'(x) = 27x^26 +25x^24 +e^(x^3)* 3(x^2)
We see that term by term, 27x^6 >0 , 25x^4 > 0.
And both factors in e^(x^3) * (3x^2) are > 0 for allx.
Therefore f'(x) > 0.
Therefore f(x) = x^27+x^25+e(x^3) is incresing funtion for all x.
We have to find the monotony of the function: f(x)=x^27+x^25+e^(x^3)
For doing this first we find the derivative of the function.
=> f'(x) = 27x^26 + 25x^24 + 2e^(x^3)*(x^2)
Now we see that for all values of x, 27x^26 + 25x^24 + 2e^(x^3)*(x^2) is always positive.
A positive first derivative of an expression in the interval (a,b) indicates that it is increasing in the interval (a,b).
Here we have the interval (a,b) as (-inf, +inf).
Therefore the function is increasing for all values of x.