The function f(x) = 2 - x³ - x⁵  is decreasing for

Concept:

• If f′(x) > 0 then the function is said to be strictly increasing.
• If f′(x) < 0 then the function is said to be decreasing.

Calculation:

Given: f(x) = 2 - x3 - x⁵

Differentiating with respect to x, we get

⇒ f'(x) = 0 - 3x² - 5x⁴

⇒ f'(x) =  - 3x2 - 5x4

For decreasing function, f'(x) < 0

⇒- 3x2 - 5x4 < 0

⇒ -[x²(3 + 5x²)] < 0

As we know, Multiplying/Dividing each side of an inequality by a negative number reverses the direction of the inequality symbol.

⇒ x2(3 + 5x2) > 0

As we know, x² ≥ 0,  x ∈ R

So, 3 + 5x² > 0, x ∈ R

Hence function is decreasing for all values of x