Question
Question: The exhaustive set of values of \[m\] for which \[f(x) = \dfrac{{{x^3}}}{3} + (m - 1){x^2} + (m + 5)...
The exhaustive set of values of m for which f(x)=3x3+(m−1)x2+(m+5)x+7 is increasing function ∀x>0 is [K,∞) then the value of K is
Solution
Hint : The derivative of a function may be used to determine whether the function is increasing or decreasing on any intervals in its domain. If f′(x)>0 at each point in an interval I, then the function is said to be increasing on I. f′(x)<0 at each point in an interval I, then the function is said to be decreasing on I.
Complete step-by-step answer :
If f′(x)>0 then f is increasing on the interval, and if f′(x)<0 then f is decreasing on the interval.
The steps involved in the process of finding the intervals of increasing and decreasing function, are as follows:
Firstly, differentiate the given function with respect to the constant variable.
Then solve f′(x)=0 .
After solving the equation of the first derivative and finding the points of discontinuity we get the open intervals with the value of x , through which the sign of the intervals can be taken into consideration.
If the sign of the interval in their first derivative form gives more than 0 then the function is said to be increasing in nature, while if the sign of the intervals in their first derivative form gives less than 0 then the function is said to be decreasing in nature.
Finally, we get increasing as well as decreasing intervals of the function.
We are given the function f(x)=3x3+(m−1)x2+(m+5)x+7
On differentiating the function with respect to x we get ,
f′(x)=x2+2(m−1)x+(m+5)
Given f(x) is increasing ∀x>0
m−1>0 and m+5>0
Therefore m>1
Hence we get K=1 .
So, the correct answer is “K = 1”.
Note : In determining intervals where a function is increasing or decreasing, you first find domain values where all critical points will occur; then, test all intervals in the domain of the function of these values to determine if the derivative is positive or negative.