# Give exact answers in this part of the question.

# Question

The temperature *g *(*t*) at time *t* of a given point of a heated iron rod is given by

*g *(*t*) = , where *t* > 0.

(a) Find the interval where *g**¢ *(*t*) > 0.

(4)

(b) Find the interval where *g**² *(*t*) > 0 and the interval where *g**² *(*t*) < 0.

(5)

(c) Find the value of *t *where the graph of *g *(*t*) has a point of inflexion.

(3)

(d) Let *t** be a value of *t* for which *g**¢ *(*t**) = 0 and *g**² *(*t**) < 0. Find *t**.

(3)

(e) Find the point where the normal to the graph of *g *(*t*) at the point

(*t**, *g *(*t**)) meets the *t*-axis.

(3)

(Total 18 marks)

# Answer

Let *f *(*x*)* = *ln |*x*5 – 3*x*2|, –0.5 <* x < *2*, x **¹ a, x **¹ b;* (*a, b *are values of* x *for which *f *(*x*)* *is not defined).

(a) (i) Sketch the graph of *f *(*x*), indicating on your sketch the number of zeros of* f *(*x*)*. *Show also the position of any asymptotes.

(2)

(ii) Find all the zeros of* f *(*x*), (that is, solve *f *(*x*) = 0).

(3)

(b) Find the **exact **values of *a *and *b.*

(3)

(c) Find* f *(*x*), and indicate clearly where *f**¢ *(*x*)* *is not defined.

(3)

(d) Find the **exact **value of the* x*-coordinate of the local maximum of* f *(*x*), for 0 <* x < *1.5. (You may assume that there is no point of inflexion.)

(3)

(e) **Write down **the definite integral that represents the area of the region **enclosed **by* f *(*x*)* *and the* x*-axis. (Do **not **evaluate the integral.)