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.