[Maximum number: 3]

The function f is defined by $\mathrm{f}(x)=\frac{1}{3 x+2}+x^{2}$ for x<-1.
Determine whether f is an increasing function, a decreasing function or neither.

[Maximum number: 4]

A curve has equation $y=2 x^{\frac{3}{2}}-3 x-4 x^{\frac{1}{2}}+4$ . Find the equation of the tangent to the curve at the point (4,0).

[Maximum number: 3]

It is given that $\mathrm{f}(x)=(2 x-5)^{3}+x$ , for $x \in \mathbb{R}$ . Show that f is an increasing function.

[Maximum number: 3]

Find the gradient of the curve

y=3 e^{4 x}-6 \ln (2 x+3)

at the point for which x=0.

The equation of a curve is $y=\frac{1+x}{1+2 x}$ for $x>-\frac{1}{2}$ . Show that the gradient of the curve is always negative.

[Maximum number: 4]

The function f is defined by $\mathrm{f}(x)=\frac{1}{3}(2 x-1)^{\frac{3}{2}}-2 x$ for $\frac{1}{2}<x<a$ . It is given that f is a decreasing function.

Find the maximum possible value of the constant a.

[Maximum number: 4]

The function f is defined by $\mathrm{f}(x)=x^{3}+2 x^{2}-4 x+7$ for $x \geqslant-2$ . Determine, showing all necessary working, whether f is an increasing function, a decreasing function or neither.

[Maximum number: 4]

A function f is defined by $\mathrm{f}: x \mapsto x^{3}-x^{2}-8 x+5$ for x<a. It is given that f is an increasing function. Find the largest possible value of the constant a.

[Maximum number: 3]

It is given that $\mathrm{f}(x)=\frac{1}{x^{3}}-x^{3}$ , for x>0. Show that f is a decreasing function.

[Maximum number: 2]

The equation of a curve is $y=4 \sqrt{ } x+\frac{2}{\sqrt{ } x}$ .

(a)

Obtain an expression for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ .

(b)

A point is moving along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.12 units per second. Find the rate of change of the y-coordinate when x=4.

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