CUET UG 2025 — Mathematics Calculus

The function $f(x) = 6 - 6x - 2x^2$f(x)=6−6x−2x2

To find where the function increases or decreases, find the derivative:

$$f'(x) = -6 - 4x$$f′(x)=−6−4x

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The critical point occurs when $f'(x) = 0$f′(x)=0:

$$-6 - 4x = 0$$−6−4x=0

$$-4x = 6$$−4x=6

$$x = -\frac{3}{2}$$x=−23​

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For $x < -\frac{3}{2}$x<−23​, test $x = -2$x=−2:

$$f'(-2) = -6 - 4(-2)$$f′(−2)=−6−4(−2)

$$f'(-2) = -6 + 8$$f′(−2)=−6+8

$$f'(-2) = 2 > 0$$f′(−2)=2>0

The function is increasing when $x < -\frac{3}{2}$x<−23​.

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For $x > -\frac{3}{2}$x>−23​, test $x = 0$x=0:

$$f'(0) = -6 - 4(0)$$f′(0)=−6−4(0)

$$f'(0) = -6 < 0$$f′(0)=−6<0

The function is decreasing when $x > -\frac{3}{2}$x>−23​.

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Therefore, the function is strictly decreasing for $x > -\frac{3}{2}$x>−23​.