 Hi and welcome to the session I am Deepika Hing. Let's discuss the question, find the intervals in which the function left given by fx is equal to 2x cube minus 3x square minus 36x plus 7 is a strictly increasing b strictly decreasing. Let us first understand the definition of strictly increasing function and strictly decreasing function. A function is strictly increasing or decreasing on an open interval where its derivative is positive or negative. That is a function is strictly increasing on an open interval where its derivative is positive and a function is strictly decreasing on an open interval where its derivative is negative. So this is a key idea behind our question. Let's start the solution given is equal to 2x cube minus 3x square minus 36x plus 7 therefore f dash x is equal to 6x square minus 6x minus 36 and this is equal to 6 we will take common x square minus x minus 6 and this is again equal to 6x square minus 3x plus 2x minus 6 and this is equal to 6x into x minus 3 plus 2 into x minus 3 that is this is equal to 6x plus 2 and x minus 3. Now f dash x is equal to 0 implies x is equal to minus 2 or 3. Now the points x is equal to minus 2 and 3 divides the real line into three disjoint intervals minus infinity to minus 2 minus 2 to 3 and 3 to infinity. Now consider when x is less than minus 2 or x plus 2 and x minus 3 both will be negative x plus 2 and x minus 3 both are negative therefore f dash x is equal to 6 into x plus 2 into x minus 3 and this will be positive therefore f dash x is increasing in the interval minus infinity to 2. Now consider the interval when minus 2 is less than x x is less than 3 then plus 2 is positive and x minus 3 will be negative therefore f dash x is equal to 6 into x plus 2 into x minus 3 is negative this implies f dash x is less than 0 and this implies f is strictly decreasing in the open interval minus 2 to 3. This is greater than 3 x minus 3 and x plus 2 both are positive therefore f dash x is equal to which is 6 into x plus 2 into x minus 3 is positive this implies f dash x is greater than 0 hence f is strictly increasing in the open interval 3 to infinity hence the answer for part a that is the given function fx is strictly increasing in the interval minus infinity to minus 2 and 3 to infinity. So this is the answer for part a f is strictly decreasing in an open interval minus 2 to 3 and this is the answer for part b I hope the question is clear to you bye and take care