Question

Write the interval in which the
function f(x) = x², is strictly
increasing
​

Answer 1

Given Question :-

• Write the interval in which the function f(x) = x², is strictly increasing.

Answer

Given :-

• A function f(x) = x²,

To Find :-

• Intervals of strictly increasing.

Concept :-

Definition :-

• A function is said to be an increasing function if the value of y increases with the increase in x.

• A function is said to be a decreasing function if the value of y decreases with the increase in x. 

We can use the first derivative test to check whether the function is increasing or decreasing.

Let f be continuous on [a, b] and differentiable on the open interval (a, b). Then

• (a) If f′(x) > 0 for each x ∈ (a, b) then f is strictly increasing in interval (a, b)
• (b) If f′(x) < 0 for each x ∈ (a, b) then f is strictly decreasing  in interval (a, b)
• (c) If f′(x) = 0 for each x ∈ (a, b) then f is a constant function in (a, b)

Let's Solve the problem now!!

CALCULATION :-

Given,

$\rm :\longmapsto\:f(x) = {x}^{2}$

• Differentiate both sides w. r. t. x, we get

$\rm :\longmapsto\:\dfrac{d}{dx} f(x) = \dfrac{d}{dx} {x}^{2}$

$\rm :\longmapsto\:f'(x) = 2x$

Now,

• For f(x) to be strictly increasing,

$\rm :\longmapsto\:f'(x) > 0$

$\rm :\implies\:2x > 0$

$\rm :\implies\:x > 0$

$\rm :\implies\: \boxed{ \bf \: x \: \in \: (0, \: \infty)}$

Additional Information :-

Properties of Monotonic Functions

Increasing and decreasing functions have certain algebraic properties, which may be useful in the investigation of functions. Here are some of them:

• If the functions f and g are increasing (decreasing) on the interval (a,b), then the sum of the functions f+g is also increasing (decreasing) on this interval.

• If the function f is increasing (decreasing) on the interval (a,b), then the opposite function −f is decreasing (increasing) on this interval.

• If the function f is increasing (decreasing) on the interval (a,b), then the inverse function 1f is decreasing (increasing) on this interval.

• If the functions f and g are increasing (decreasing) on the interval (a,b) and moreover, f≥0, g≥0, then the product of the functions fg is also increasing (decreasing) on this interval.