1.5.6 Polynomials and rational expressions

The curve $C$ has equation

2x^3 + \frac{18}{x} + 5, \quad x > 0

a.

Find $dydx\frac{\mathrm{d}y}{\mathrm{d}x}$ , giving your answer in simplest form.

[2]

b.

Given that

the point $P$ lies on $C$
the line with equation $y = k - 24 x$ , where $k$ is a constant, is the tangent to $C$ at $P$

Show that the $x$ -coordinate of $P$ satisfies the equation

x^4 + 4x^2 - 3 = 0

[3]

c.

Hence find the exact value of $k$ .

[4]

The curve $C$ has equation

2x^3 + \frac{18}{x} + 5, \quad x > 0

a.

Find $dydx\frac{\mathrm{d}y}{\mathrm{d}x}$ , giving your answer in simplest form.

[2]

b.

Given that

the point $P$ lies on $C$
the line with equation $y = k - 24 x$ , where $k$ is a constant, is the tangent to $C$ at $P$

Show that the $x$ -coordinate of $P$ satisfies the equation

x^4 + 4x^2 - 3 = 0

[3]

c.

Hence find the exact value of $k$ .

[4]

1.5.6 Polynomials and rational expressions Questions