Let f ,(x) = 1 + 2 x^2 + 2^2 x^4 + ..... + 2^ 10 x^ 20 , then f(x… - Vidyadip

MCQ

Let $f\,(x) = 1 + 2{x^2} + {2^2}{x^4} + ..... + {2^{10}}{x^{20}}$, then $f(x)$ has

A
More than one minimum
✓
Exactly one minimum
C
At least one maximum
D
None of these

✓

Answer

Correct option: B.

Exactly one minimum

b
(b) $f(x) = 1 + 2{x^2} + {2^2}{x^4} + {2^3}{x^6} + {...2^{10}}{x^{20}}$

$f'(x) = x(4 + {4.2^2}{x^2} + .... + {20.2^{10}}{x^{18}})$

$\therefore $ $f'(x) = 0$ ==> $x = 0$ only.

Also $f''(0) > 0$.

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