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Question
Any tangent to the curve y = 2x7 + 3x + 5 __________________ .
Options
is parallel to x-axis
is parallel to y-axis
makes an acute angle with x-axis
makes an obtuse angle with x-axis
Solution
makes an acute angle with x-axis
We have, y = 2x7 + 3x + 5
\[\frac{dy}{dx} = 14 x^6 + 3\]
\[ \Rightarrow \frac{dy}{dx} > 3 \left( \because x^6 \text { is always positive for any real value of } x \right)\]
\[ \Rightarrow \frac{dy}{dx} > 0\]
\[\text { So }, \tan\theta > 0\]
\[\text { Hence }, \theta \text { lies in first quadrant } . \]
\[\text { Thus, the tangent to the curve makes an acute angle with x - axis }\]
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