The interval on which the function f(x) = x^3 + 2x^2 - 1 is decreasing, is Option 1: (-∞, -4/3) Option 2: [0, ∞) Option 3: [-4/3, 0) Option 4: [-4/3, ∞)

Answer: [-4/3, 0). Solution: Given: f(x) = x^3 + 2x^2 - 1 A function is decreasing when its derivative is negative. Finding the derivative using the power rule: f'(x) = 3x^2 + 4x To find critical points, set f'(x) = 0: 3x^2 + 4x = 0 x(3x + 4) = 0 This gives x = 0 or 3x + 4 = 0 Therefore x = 0 or x = -4/3 Testing the sign of f'(x) = x(3x + 4) in each interval: For x 0 For -4/3 0, test x = 1: f'(1) = (1)(3(1) + 4) = (1)(7) = 7 > 0 The function is decreasing where f'(x) < 0, which occurs in the interval -4/3 < x < 0. In interval notation: [-4/3, 0) The bracket at -4/3 includes the endpoint, while the parenthesis at 0 excludes it. Therefore, the function is decreasing on [-4/3, 0).

Calculus

Application of Derivatives

Medium

common

The interval on which the function $f(x) = x^3 + 2x^2 - 1$ is decreasing, is

$(-\infty, -4/3)$

$[0, \infty)$

$[-4/3, 0)$

$[-4/3, \infty)$

✅ Correct Option: 3

Given: $f(x) = x^3 + 2x^2 - 1$

A function is decreasing when its derivative is negative.

Finding the derivative using the power rule:

$f'(x) = 3x^2 + 4x$

To find critical points, set $f'(x) = 0$ :

$3x^2 + 4x = 0$

$x(3x + 4) = 0$

This gives $x = 0$ or $3x + 4 = 0$

Therefore $x = 0$ or $x = -\dfrac{4}{3}$

Testing the sign of $f'(x) = x(3x + 4)$ in each interval:

For $x < -\dfrac{4}{3}$ , test $x = -2$ :

$f'(-2) = (-2)(3(-2) + 4) = (-2)(-2) = 4 > 0$

For $-\dfrac{4}{3} < x < 0$ , test $x = -1$ :

$f'(-1) = (-1)(3(-1) + 4) = (-1)(1) = -1 < 0$

For $x > 0$ , test $x = 1$ :

$f'(1) = (1)(3(1) + 4) = (1)(7) = 7 > 0$

The function is decreasing where $f'(x) < 0$ , which occurs in the interval $-\dfrac{4}{3} < x < 0$ .

In interval notation: $\left[-\dfrac{4}{3}, 0\right)$

The bracket at $-\dfrac{4}{3}$ includes the endpoint, while the parenthesis at $0$ excludes it.

Therefore, the function is decreasing on $\left[-\dfrac{4}{3}, 0\right)$ .

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The interval on which the function f(x)=x3+2x2−1f(x) = x^3 + 2x^2 - 1f(x)=x3+2x2−1 is decreasing, is

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CUET Mathematics Past Year Questions (Topic-Wise)

The interval on which the function $f(x) = x^3 + 2x^2 - 1$ is decreasing, is