QJEEM08Ma6bdc: JEE Main 2008 Mathematics - Class 12 JEEM08Ma6bdc Solution: How many real solutions does the equation x 7 + 14 x 5 + 16 x 3 …

Question

Verified Solution • 2025–26 Session moderate

How many real solutions does the equation

x^{7} + 14 x^{5} + 16 x^{3} + 30 x - 560 = 0

have?

Accepted Answer

Let

f (x) = x^{7} + 14 x^{5} + 16 x^{3} + 30 x - 560

\Rightarrow f^{'} (x) = 7 x^{6} + 70 x^{4} + 48 x^{2} + 30 > 0, \forall x \in R

\Rightarrow f

is an increasing function on

R

Also

lim_{x \to \infty} f (x) = \infty

and

lim_{x \to - \infty} f (x) = - \infty

\Rightarrow

The curve

y = f (x)

crosses

x

-axis only once.

∴

f (x) = 0

has exactly one real root.

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