If is a monotonically decreasing function of in the largest possible interval , then the value if is
A)
B)
C)
D) None of these
Answer
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Hint:
The given function is monotonically decreasing in the given interval, which means that its derivative is negative in this interval. So, for solving this question, we will first differentiate the given function. Since the given interval is the largest possible interval in which the given function can monotonically decrease, this means that the derivative at the end points must be equal to zero. Therefore, on equating the derivative of the function equal to zero at any of the end points, we will get the required value of .
Complete step by step solution:
The function given in the question is .
Differentiating both sides with respect to , we get
……………………….
According to the question, the given function is monotonically decreasing in the interval . This means that the slope of is negative in this interval. Further, it is also given that the interval is the largest possible interval in which the given function can monotonically decrease. Therefore, the slope, or the derivative of the given function is equal to zero at the end points of the given interval, which means that
………………………….
Substituting in equation , we get
Now substituting equation in above equation, we get
Adding on both the sides, we finally get
Hence, the correct answer is option A.
Note:
Although we used the left end point of the given interval to get the value of , but we can also use the right end point at which the derivative to zero. On equating the derivative to zero at also, we will get the same value of . Differentiation is a way of finding the derivative of a function or the rate of change of a function with respect to a particular variable. Integration is the inverse of differentiation and hence it is called antiderivative.
The given function is monotonically decreasing in the given interval, which means that its derivative is negative in this interval. So, for solving this question, we will first differentiate the given function. Since the given interval is the largest possible interval in which the given function can monotonically decrease, this means that the derivative at the end points must be equal to zero. Therefore, on equating the derivative of the function equal to zero at any of the end points, we will get the required value of
Complete step by step solution:
The function given in the question is
Differentiating both sides with respect to
According to the question, the given function
Substituting
Now substituting equation
Adding
Hence, the correct answer is option A.
Note:
Although we used the left end point
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