Question
Question: If \[f(x)={{x}^{5}}-20{{x}^{3}}+240x,\] then \( f(x) \) satisfies which of the following A. It is ...
If f(x)=x5−20x3+240x, then f(x) satisfies which of the following
A. It is monotonically decreasing everywhere
B. It is monotonically decreasing only in (0,∞)
C. It is monotonically increasing everywhere
D. It is monotonically increasing only in (−∞,0)
Solution
We will use the concept of differentiation to find whether it is monotonically increasing or decreasing and in what range. According to the concept, if the first derivative of a function gives zero then there will be an interval for increasing and decreasing and if the first derivative is not zero then the function may be either monotonically increasing or decreasing.
Complete step by step answer:
Moving ahead with the question in step wise manner;
We are asked to find whether the function is monotonically increasing or decreasing and in what range. Since according to the concept of differentiation we know that if the first derivative of function gives zero then at that point function will change its character from increasing to decreasing or decreasing to increasing, i.e. if the function will be decreasing before the point it give its first derivative zero then after that point it will be continuously increasing till the next point comes where the first derivative is zero.
And if the first derivative of a function does not come zero then the function is either continuously increasing or continuously increasing everywhere. To find whether function is continuously increasing or decreasing we can come to point by seeing its character i.e. for a function to be increasing the function at some point ‘a’ should be greater than some point ‘b’ if point b>a , as it can be denotes as for continuously increasing function f(b)>f(a);b>a .
Now for our case let us first find the points at which it gives zero. For that let us first find the first derivative, i.e. dxdf(x) , which will be equal to;