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Question

The maximum value of f(x)=[x(1)+1]13,0x1 is\\
a) (13)13
b) 12
c) 1
d) zero

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Solution

Let f(x)=[x(1)+2]13,0x1
=(x2x+1)13
On differentiating w.r.t x, we get
f(x)=13(x2x+1)131(2x1)=1(2x1)3(x2x+1)23
Now, put f(x)=02x1=0x=12ϵ[0,1]
So, x=12 is a critical point.

Now, we evaluate the value of f at critical point x=12 and at the end points of the interval [0,1].
At x=0f(0)=(00+1)13=1At x=1f(1)=(11+1)13=1At x=12,f(12)=(1412+1)13=(34)13
Maximum value of f(x) is 1 at x=0, 1.
Hence, (c) is the correct option.


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