Question

Tangent to y=5x⁵+10x+15......,Select correct option from the given options.
(a) is always vertical
(b) is always horizontal
(c) makes acute angle with the positive X-axis
(d) makes obtuse angle with the positive X-axis

Answer 1

answer : option (c) makes acute angle with positive X- axis.

explanation : function, y = 5x⁵+10x+15

differentiating y with respect to x,

dy/dx = 5 d(x⁵)/dx + 10 d(x)/dx + d(15)/dx

= 5 × 4x⁴ + 10 + 0

= 20 x⁴ + 10

here domain of y = y=5x⁵+10x+15 is all real numbers.

I.e., -∞ < x < ∞

⇒0 ≤ x⁴ < ∞

⇒0 ≤ 20x⁴ < ∞

⇒10 ≤ 20x⁴ + 10 < ∞

i.e., dy/dx ≥ 10 , it means slope of tangent of curve always will be positive.

but slope , m = tanθ

and we know, tanθ is positive in two quadrant first and 3rd.

first quadrant, [0, π/2] ⇒an acute angle will form.

but 3rd quadrant [π, 3π/2] ⇒not obtuse angle.

so, the tangent makes acute angle with the positive x-axis.