Any tangent to the curve y = 2x⁷ + 3x + 5

(a) is parallel to x-axis
(b) is parallel to y-axis
(c) makes an acute angle with x-axis
(d) makes an obtuse angle with x-axis

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Solution

(c) makes an acute angle with x-axis

We have, y = 2x⁷ + 3x + 5
$\frac{d y}{d x} = 14 x^{6} + 3 \Rightarrow \frac{d y}{d x} > 3 (∵ x^{6} is always positive for any real value of x) \Rightarrow \frac{d y}{d x} > 0 S o, \tan θ > 0 Hence, θ lies in first quadrant . Thus, the tangent to the curve makes an acute angle with x - axis$

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Any tangent to the curve y = 2x7 + 3x + 5 (a) is parallel to x-axis (b) is parallel to y-axis (c) makes an acute angle with x-axis (d) makes an obtuse angle with x-axis

(c) makes an acute angle with x-axis We have, y = 2x7 + 3x + 5 dydx=14x6+3⇒dydx>3 ∵x6 is always positive for any real value of x⇒dydx>0So, tanθ>0Hence, θ lies in first quadrant.Thus, the tangent to the curve makes an acute angle with x-axis

Any tangent to the curve y = 2x⁷ + 3x + 5

(a) is parallel to x-axis
(b) is parallel to y-axis
(c) makes an acute angle with x-axis
(d) makes an obtuse angle with x-axis