Question

A function f(x) = 10 – x – 2x² is increasing on the interval ______.

विकल्प

• `(-∞, -1/4]`

• `(-∞, 1/4)`

• `[-1/4, ∞)`

• `[-1/4, 1/4]`

Answer 1

A function f(x) = 10 – x – 2x² is increasing on the interval `underlinebb((-∞, -1/4])`.

Explanation:

Calculating f'(x):

f(x) = 10 – x – 2x² 

f'(x) = –1 – 2 × 2x

f'(x) = –(1 + 4x)

Putting f'(x) = 0

–1(1 + 4x) = 0

1 + 4x = 0

4x = –1

x = `(-1)/4`

Plotting points on a number line:

Value of x	Intervals	Sign of f'(x) = –(1 + 4x)	Nature of function f
`x < (-1)/4`	`(-∞, (-1)/4)`	(–) (–) > 0	f is strictly increasing.
`x > (-1)/4`	`((-1)/4, ∞)`	(–) (+) < 0	f is strictly decreasing.

Hence, f is strictly decreasing for `x > (-1)/4`.

f is strictly increasing for `x < (-1)/4`.

Since f is strictly increasing for `x < (-1)/4`.

Therefore, f is increasing for `x ≤ (-1)/4`.

We write f is increasing in `(-∞, -1/4]`.