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**Illustration 1: The maximum value of ****(cos a _{1}). (cos a_{2})â€¦. (cos a_{n}), under the restrictions 0 = a_{1}, a_{2},â€¦ , a_{n}â‰¤Ï€/2**

Then, cos a

Hence, cos a

and sin a

by multiplying equations (1) and (2) we get,

(cos a

Then k

Hence, k

= 1/2

Hence, k = 1/2

**Illustration 2: If ax ^{2} + bx + c = 0, a, b, c âˆˆ R. Find the condition that this equation would have at least one root in (0, 1).**

Integrating both sides,

=>f(x) = ax

=>f(0) = d and f(1) = a/3 + b/2 + c + d

Since, Rolleâ€™s theorem is applicable

=>f(0) = f(1)

=> d = a/3 + b/2 + c + d

=> 2a + 3b + 6c = 0

Hence required condition is 2a + 3b + 6c = 0

**Illustration 3: If at each point of the curve y = x ^{3} â€“ ax^{2} + x + 1 the tangents is inclined at an acute angle with the positive direction of the x-axis, then find the interval in which a lies.**

Hence,dx/dy> 0, So, 3x

=> (2a)

=> 4(a

=> (a â€“ âˆš3) (a + âˆš3) < 0

So, â€“ âˆš3 < a <âˆš3.

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