the set of all the possible values of 'a' for which the function f(x)=5+(a-2)x+(a-1)x^2-x^3 has a local minimum value at some x<1 and local maximum value at some x>1 is

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Solution

$D e a r S t u d e n t D i f f e r e n t i a t i n g i t w i l l g i v e 2 (a - 2) x - 3 x^{2} I n o r d e r f o r t h e f u n c t i o n t o h a v e c r i t i c a l p o i n t s i n r a n g e x < 1 2 (a - 2) x - 3 x^{2} = 0 \Rightarrow 2 (a - 2) - 3 x = 0 \Rightarrow 2 (a - 2) = 3 x \Rightarrow \frac{2 (a - 2)}{3} = x F o r x < 1 \frac{2 (a - 2)}{3} < 1 \Rightarrow a < 7 / 2 F o r x > 1 \frac{2 (a - 2)}{3} > 1 a > 7 / 2 R e g a r d s$

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