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# Any tangent to the curve

Question:

Any tangent to the curve $y=2 x^{7}+3 x+5$.

A. is parallel to $x$-axis

B. is parallel to $y$-axis

C. makes an acute angle with $\mathrm{x}$-axis

D. makes an obtuse angle with $x$-axis

Solution:

Given curve $y=2 x^{7}+3 x+5$

Differentiating w.r.t. $x$,

$\frac{\mathrm{dy}}{\mathrm{dx}}=14 \mathrm{x}^{6}+3$

Here $\frac{\mathrm{dy}}{\mathrm{dx}} \geq 3$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}>0$

So, $\tan \theta>0$

Hence, $\theta$ lies in first quadrant.

So, any tangent to this curve makes an acute angle with $\mathrm{x}$-axis.