**1.**
The diagram above shows a sketch of the curve *C* with the equation .
(a) Find the coordinates of the point where *C* crosses the *y*-axis.
**(1)**
(b) Show that *C* crosses the *x*-axis at *x* = 2 and find the *x*-coordinate of the other point where *C* crosses the *x*-axis.
**(3)**
(c) Find .
**(3)**
(d) Hence find the exact coordinates of the turning points of *C*.
**(5)**
**(Total 12 marks)**
**2.** (a) By writing sec *x* as show that = sec *x* tan *x*.
**(3)**
Given that *y* = *e*^{2}^{x} sec 3*x*,
(b) find
**(4)**
The curve with equation *y* = e^{2}^{x} sec 3*x*, – has a minimum turning point at (a, b).
(c) Find the values of the constants a and b, giving your answers to 3 significant figures.
**(4)**
**(Total 11 marks)**
3. A curve C has equation
y = x^{2}e^{x}.
(a) Find , using the product rule for differentiation.
**(3)**
(b) Hence find the coordinates of the turning points of C.
**(3)**
(c) Find
**(2)**
(d) Determine the nature of each turning point of the curve C.
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