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  • The points at which the tangents to the curve y = x3 - 12x + 18 are parallel to x-axis are: A. (2, -2), (-2, -34) B. (2, 34), (-2, 0) C. (0, 34), (-2, 0) D. (2, 2), (-2, 34)

The points at which the tangents to the curve y = x^3 - 12x + 18 are parallel to x-axis are:
A. (2, -2), (-2, -34)
B. (2, 34), (-2, 0)
C. (0, 34), (-2, 0)
D. (2, 2), (-2, 34)

Answers (1)

D)

Given the equation of the curve is

y = x^3 - 12x + 18

Differentiating on both sides with respect to x, we get

\frac{\mathrm{d}(\mathrm{y})}{\mathrm{dx}}=\frac{\mathrm{d}\left(\mathrm{x}^{3}-12 \mathrm{x}+18\right)}{\mathrm{dx}}
Applying the sum rule of differentiation, we get
\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}\left(\mathrm{x}^{3}\right)}{\mathrm{dx}}-12 \frac{\mathrm{d}(\mathrm{x})}{\mathrm{dx}}+\frac{\mathrm{d}(18)}{\mathrm{dx}}
We know derivative of a constant is 0,so above equation becomes
\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}\left(\mathrm{x}^{3}\right)}{\mathrm{dx}}-12 \frac{\mathrm{d}(\mathrm{x})}{\mathrm{dx}}+0
Applying the power rule we get
\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3 \mathrm{x}^{2}-12 \ldots(i)
Thus, the slope of line parallel to the x -axis is given by
\frac{\mathrm{dy}}{\mathrm{dx}}=0
So equating equation (i) to 0 we get
3 x^{2}-12=0

\\ { \Rightarrow 3x\textsuperscript{2}=12}\\ { \Rightarrow x\textsuperscript{2}=4}\\ { \Rightarrow x= \pm 2}\\

When x=2, the given equation of curve becomes,

\\ {y = x\textsuperscript{3} - 12x + 18}\\ { \Rightarrow y = (2)\textsuperscript{3} - 12(2) + 18}\\ { \Rightarrow y = 8- 24 + 18}\\ { \Rightarrow y = 2}\\

When x=-2, the given equation of curve becomes,

\\ {y = x\textsuperscript{3} - 12x + 18}\\ { \Rightarrow y = (-2)\textsuperscript{3} - 12(-2) + 18}\\ { \Rightarrow y = -8+24 + 18}\\ { \Rightarrow y = 34}\\

Hence, the points at which the tangents to the curve y = x^3 - 12x + 18 are parallel to x-axis are (2, 2) and (-2, 34).

So, the correct option is option D.

Posted by

infoexpert22

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