let `f(x)=1+2x^2+2^2x^4+.....+2^(10)x^(20).` The , `f(x)` has

Options

A. more than one minimum
B. exactly one minimum
C. atleast one maximum
D. None of the above

Answer

B

Explanation

Given `f(x)=1+2x^(2)+2^(2)x^(4)+….+2^(10)x^(20)`
`impliesf'(x)=4x+4.2^(2)x^(3)+………+20.2^(10)x^(19)`
`=x(4+4.2^(2)x^(2)+…………+20.2^(10)x^(18))`
For a maximum or minimum put `f'(x)=0`
`impliesx=0`
But, `f''(x)=4+12.2^(2)x^(2)+………+20.19.2^(10)x^(18)gt0`
At `x=0, f''(x)=4gt0`
`:.f(x)` is minimum at `x=0`
For `x lt 0impliesf'(x)lt0` and `x gt0impliesf'(x)gt0`
Hence, if has exactly one minimum value exist.