Maxima and MinimaHard
Question
If (x - a)2m (x - b)2n +1, where m and n are positive integers and a > b, is the derivative of a function f, then-
Options
A.x = a gives neither a maximum, nor a minimum
B.x = a gives a maximum
C.x = b gives neither a maximum nor a minimum
D.None of these
Solution
We have f′(x) = (x - a)2n (x - b)2m + 1
∴ f′(x) = 0 ⇒ x = a, b
when x = a - h
f′(x) = h2n (a - h - b)2m +1
when x = a + h
f′(x) = h2n (a + h - b)2m + 1
Thus, we see that as x passes through a, f′(x) does not change sign. Hence, there is neither a maximum nor a minimum at x = a
∴ f′(x) = 0 ⇒ x = a, b
when x = a - h
f′(x) = h2n (a - h - b)2m +1
when x = a + h
f′(x) = h2n (a + h - b)2m + 1
Thus, we see that as x passes through a, f′(x) does not change sign. Hence, there is neither a maximum nor a minimum at x = a
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