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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 mm for which f(x)=x33+(m1)x2+(m+5)x+7f(x) = \dfrac{{{x^3}}}{3} + (m - 1){x^2} + (m + 5)x + 7 is increasing function x>0\forall x > 0 is [K,)[K,\infty ) then the value of KK is

Explanation

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)>0f'(x) > 0 at each point in an interval I, then the function is said to be increasing on I. f(x)<0f'(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)>0f'(x) > 0 then ff is increasing on the interval, and if f(x)<0f'(x) < 0 then ff 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)=0f'(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 xx , 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 00 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 00 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)=x33+(m1)x2+(m+5)x+7f(x) = \dfrac{{{x^3}}}{3} + (m - 1){x^2} + (m + 5)x + 7
On differentiating the function with respect to xx we get ,
f(x)=x2+2(m1)x+(m+5)f'(x) = {x^2} + 2(m - 1)x + (m + 5)
Given f(x)f(x) is increasing x>0\forall x > 0
m1>0m - 1 > 0 and m+5>0m + 5 > 0
Therefore m>1m > 1
Hence we get K=1K = 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.