Find the points on the curve $y = x^3 - 3x^2 - 4x$ at which the tangents are parallel to the line $4x + y - 3 = 0$.

Find the points on the curve y = x³ - 3x² - 4x at which — CBSE Class 12 Mathematics 2020 PYQ

Question

y = x^3 - 3x^2 - 4x

CBSECBSE Class XII Board 2020Subjective· 2mImportance★★★★★

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Concept understanding — Derivative Evaluation

Derivative Evaluation

To evaluate a derivative means to find $f'(a)$ — a single number that tells you how fast $f$ is changing right at $x=a$ . Think of a speedometer: it doesn't report how far you've travelled, only how fast your position is changing at this instant. That instantaneous rate of change is exactly what $f'(a)$ measures.

The geometric picture

On the curve $y=f(x)$ , pick a point $P$ and a nearby point $Q$ . The straight line through them — the secant — has slope equal to the average rate of change between $P$ and $Q$ . Now slide $Q$ toward $P$ : the secant rotates into the tangent line that just touches the curve at $P$ , and its slope is $f'(a)$ .

Note

$f'(a)$ is the slope of the tangent to $y=f(x)$ at $x=a$ — how steep the curve is right there.

The limit definition

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

Here $h$ is a tiny step from $a$ to $a+h$ , the numerator is the matching change in height, and the ratio is a secant slope. As $h\to 0$ the secant becomes the tangent. An equivalent form is

f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}.

When this limit exists, $f$ is differentiable at $a$ (which forces continuity there).

Watch out

Continuity alone is not enough. $f(x)=|x|$ is continuous at $0$ , but its left slope $-1$ and right slope $+1$ disagree, so $f'(0)$ does not exist — a corner has no single tangent.

A worked evaluation

For $f(x)=x^2$ at $x=3$ :

f'(3) = \lim_{h \to 0} \frac{(3+h)^2 - 9}{h} = \lim_{h \to 0} (6 + h) = 6.

So the tangent at $x=3$ has slope $6$ .

From a number to a function …

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The key idea is that parallel lines have equal slopes. The slope of the given line is $-4$ , so we set the derivative of the curve equal to $-4$ , solve for $x$ , and then find the corresponding $y$ coordinates. The required points are $(3, -12)$ and $(-1, 0)$ .

Concept and Intuition

When two lines are parallel, they have the same slope. Here, we are given a fixed line $4x + y - 3 = 0$ , and we want points on the cubic curve where the tangent line has that same slope.

The slope of a tangent to a curve at any point is given by the derivative $\frac{dy}{dx}$ evaluated at that point. So the problem reduces to: find all $x$ such that $\frac{dy}{dx} = -4$ , then find the corresponding $y$ from the curve equation.

Step-by-step solution

1. Find the slope of the given line.

Rewrite $4x + y - 3 = 0$ as $y = -4x + 3$ .

The coefficient of $x$ is $-4$ , so the slope of this line is $m = -4$ .

Note

Any line parallel to this one must also have slope $-4$ .

2. Differentiate the curve equation.

We have $y = x^3 - 3x^2 - 4x$ .

Differentiating term by term:

\frac{dy}{dx} = 3x^2 - 6x - 4

3. Set the derivative equal to the required slope.

We want the tangent slope to be $-4$ , so:

3x^2 - 6x - 4 = -4

4. Solve for $x$ .

Add $4$ to both sides:

3x^2 - 6x = 0

Factor:

3x(x - 2) = 0

So $x = 0$ or $x = 2$ .

Watch out

A common mistake is to stop here and give only the $x$ -coordinates. The question asks for points, so we must also find the $y$ -coordinates.

5. Find the corresponding $y$ -coordinates.

Substitute each $x$ into the original curve equation $y = x^3 - 3x^2 - 4x$ . …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 103 on this concept.

CBSE 2026Set 65/3/11 markMCQ
$\sin^{-1} x = y$
›Reveal solutionSolution
The derivative of $\sin^{-1} x$ is found by implicit differentiation of $x = \sin y$ , giving $\frac{dy}{dx} = \frac{1}{\cos y} = \frac{1}{\sqrt{1-x^2}}$ , which matches option (B) $\cos y$ only if we interpret it as $\sec y$ — but careful: the correct form is $\frac{1}{\sqrt{1-x^2}}$ , and among the given choices, (B) $\cos y$ is actually $\frac{1}{\cos y}$ ? No — let's check: $\cos y = \sqrt{1-x^2}$ , so $\frac{1}{\cos y} = \sec y$ , which is option (D). The final answer is (D) $\sec y$ .

The core idea: when you have an inverse trigonometric function, the easiest way to differentiate it is to rewrite it as a direct trigonometric equation and then use implicit differentiation. This avoids memorising a dozen formulas and builds from what you already know — the derivative of $\sin$ and the chain rule.

Let $\sin^{-1} x = y$ . This means $x = \sin y$ , and importantly, $y$ is restricted to $[-\pi/2, \pi/2]$ so that $\cos y \ge 0$ .

Start with the relation:

$x = \sin y$

Differentiate both sides with respect to $x$ . Remember $y$ is a function of $x$ , so we use the chain rule on the right:

$\frac{d}{dx}(x) = \frac{d}{dx}(\sin y)$ $1 = \cos y \cdot \frac{dy}{dx}$

Solve for $\frac{dy}{dx}$ :

$\frac{dy}{dx} = \frac{1}{\cos y}$

Now, $\cos y$ can be expressed in terms of $x$ . Since $\sin y = x$ , we use the identity $\sin^2 y + \cos^2 y = 1$ :

$\cos^2 y = 1 - \sin^2 y = 1 - x^2$ $\cos y = \sqrt{1 - x^2} \quad (\text{positive because } y \in [-\pi/2, \pi/2])$

Therefore: $\frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}}$ …
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CBSE 2026Set V11 markMCQ
$x - y = \pi$
(a) $\pi$
(b) $-\pi$
(c) $1$
(d) $-1$
›Reveal solutionSolution

Differentiating the constant-difference relation gives $\dfrac{dy}{dx}=1$ ; answer (c).

Differentiate $x-y=\pi$ (a constant) with respect to $x$ : …
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CBSE 2026Set A1 markMCQ
$\frac{d}{dx}(\log x^n) =$
(a) $\frac{1}{x^n}$
(b) $n$
(c) $\frac{1}{x}$
(d) $\frac{n}{x}$
›Reveal solutionSolution

$\frac{d}{dx}(\log x^n) = \frac{n}{x}$ .

First simplify with the log power rule:

$\log x^n = n\log x.$

Differentiate: …

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CBSE 2026Set A1 markMCQ
$\frac{d}{dx}(e^{x-a}) =$
(a) $e^{x-a}$
(b) $(x-a)e^{x-a}$
(c) $e^x$
(d) $-e^{x-a}$
›Reveal solutionSolution

$\frac{d}{dx}\left(e^{x-a}\right) = e^{x-a}$ .

Here $a$ is a constant. Let $u = x-a$ , so $\frac{du}{dx} = 1$ .

By the chain rule, …

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CBSE 2026Set A1 markMCQ
$\frac{d}{dx}(\tan^{-1}\sqrt{x} + \cot^{-1}\sqrt{x}) =$
(a) $\frac{\pi}{2}$
(b) $0$
(c) $1$
(d) $\pi$
›Reveal solutionSolution

The derivative is $0$ because the sum is a constant.

For any real argument $t$ , $\tan^{-1}t + \cot^{-1}t = \frac{\pi}{2}$ . Taking $t=\sqrt{x}$ ,

$\tan^{-1}\sqrt{x} + \cot^{-1}\sqrt{x} = \frac{\pi}{2}.$

…

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CBSE 2026Set A1 markMCQ
$\frac{d}{dx}(2\tan^{-1}x) =$
(a) $\frac{1}{1+x^2}$
(b) $\frac{2}{1+x^2}$
(c) $\frac{1}{2(1+x^2)}$
(d) $\frac{1}{1-x^2}$
›Reveal solutionSolution

$\frac{d}{dx}(2\tan^{-1}x) = \frac{2}{1+x^2}$ .

Using the standard derivative $\frac{d}{dx}\tan^{-1}x = \frac{1}{1+x^2}$ and the constant multiple rule: …

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CBSE 2026Set A1 markMCQ
$\frac{d}{dx}\left\{\lim_{x\to 0}\frac{x^5-a^5}{x-a}\right\} =$
(a) $a$
(b) $0$
(c) $5a^4$
(d) $5$
›Reveal solutionSolution

The inner limit is a constant, so the derivative is $0$ .

Evaluate the limit first (it does not depend on $x$ after the limit is taken):

$\lim_{x\to a}\frac{x^5-a^5}{x-a} = 5a^4$ …

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CBSE 2026Set A1 markMCQ
$\frac{d}{dx}(\cot^{-1}x) =$
(a) $\frac{1}{1+x^2}$
(b) $\frac{-1}{1+x^2}$
(c) $\frac{1}{x}$
(d) $\frac{-1}{x}$
›Reveal solutionSolution

$\frac{d}{dx}\cot^{-1}x = \frac{-1}{1+x^2}$ .

This is a standard result. From $\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}$ , differentiating gives …

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CBSE 2026Set A1 markMCQ
$\frac{d}{dx}\sin^{-1}(3x - 4x^3) =$
(a) $\frac{3}{\sqrt{1-x^2}}$
(b) $\frac{-3}{\sqrt{1-x^2}}$
(c) $\frac{1}{\sqrt{1-x^2}}$
(d) $\frac{-1}{\sqrt{1-x^2}}$
›Reveal solutionSolution

$\frac{d}{dx}\sin^{-1}(3x-4x^3) = \frac{3}{\sqrt{1-x^2}}$ .

Use the identity (for the principal branch, $-\frac{1}{2}\le x\le\frac{1}{2}$ ):

$\sin^{-1}(3x - 4x^3) = 3\sin^{-1}x,$

which follows from $\sin 3\theta = 3\sin\theta - 4\sin^3\theta$ with $x=\sin\theta$ .

Differentiate: …

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CBSE 2026Set A1 markMCQ
$\frac{d}{dx}(\sin 3x \cdot \cos 5x) =$
(a) $4\cos 8x$
(b) $4\cos 8x - \cos 2x$
(c) $\cos 2x$
(d) $\cos 2x - 4\cos 8x$
›Reveal solutionSolution

Convert the product to a sum, then differentiate: $\frac{d}{dx}(\sin 3x\cos 5x)=4\cos 8x-\cos 2x$ .

Use the product-to-sum identity $\sin A\cos B=\tfrac12[\sin(A+B)+\sin(A-B)]$ .

$\sin 3x\cos 5x=\tfrac12[\sin 8x+\sin(-2x)]=\tfrac12[\sin 8x-\sin 2x]$ .

Differentiate term by term:

…

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CBSE 2026Set ANNUAL1 mark
$y = \frac{e^x}{\sin x}$
›Reveal solutionSolution

Apply the quotient rule to $y=e^x/\sin x$ .

$\dfrac{dy}{dx} = \dfrac{e^x\sin x - e^x\cos x}{\sin^2 x} = \dfrac{e^x(\sin x-\cos x)}{\sin^2x}$ .

…

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CBSE 2026Set ANNUAL1 markMCQ
$\dfrac{d}{dx}(a^x) =$
(a) $a^x \log a$
(b) $\log a$
(c) $a^x$
(d) None of these
›Reveal solutionSolution

$\dfrac{d}{dx}a^x = a^x \ln a$ , using $a^x = e^{x\ln a}$ .

…

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Q.Find the points on the curve y=x3−3x2−4xy = x^3 - 3x^2 - 4xy=x3−3x2−4x at which the tangents are parallel to the line 4x+y−3=04x + y - 3 = 04x+y−3=0.

Concept and Intuition

Step-by-step solution

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Q.Find the points on the curve $y = x^3 - 3x^2 - 4x$ at which the tangents are parallel to the line $4x + y - 3 = 0$ .

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