Discussion

Explanation:

(x + 4)(x + 6)(x + 8) ⋯ (x + 98) < 0

Critical points are -98, -96, -94, … -8. -6 and -4

For x > -4, the given expression will be positive. Hence, we will reject all values of x > -4.

For -6 < x < -4, the given expression will be negative. The only integral value for x in this range is -5.

For -8 < x < -6, the given expression will be positive.

For -10 < x < -8, the given expression will be negative. The only integral value for x in this range is -9.

The given expression will be negative for x = -5, -9, -13, and so on

Now the only integral value of x between -98 and -96 is -97
∴ f(-97) = -93 × - 91 × -89 × … × -1 × 1

Here we have product of 47 negative terms and 1 positive term, hence the product will be negative.

∴ For -98 < x < -96, the given expression will be negative. The only integral value for x in this range is -97.

Finally, for x < -98, f(x) is product of 48 negative terms, which will give us a positive number. Hence, we will reject all value of x < -98.

The given expression will be negative for x = -5, -9, -13, and so on till -97

Number of possible values of x for which f(x) < 0 = (-5 + 97)/4 + 1 = 24

Hence, option (d).

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