The following function is defined over the interval [-L, L]:

f(x) = px4 + qx5.

If it is expressed as a Fourier series, 

$\rm f(x)=a_0 +\displaystyle\sum^\infty_{n=1} \left\{a_n \sin\left( \frac{\pi x}{L} \right) +b_n\cos\left( \frac{\pi x}{L} \right) \right\} $,

which options amongst the following are true?

Consider the polynomial f(x) = x³ $$-$$ 6x² + 11x $$-$$ 6 on the domain S, given by 1 $$\le$$ x $$\le$$ 3. The first and second derivatives are f'(x) and f''(x).

Consider the following statements :

I. The given polynomial is zero at the boundary points x = 1 and x = 3.

II. There exists one local maxima of f(x) within the domain S.

III. The second derivative f''(x) > 0 throughout the domains S.

IV. There exists one local minima f(x) within the domain S.

$$\int {\left( {x - {{{x^2}} \over 2} + {{{x^3}} \over 3} - {{{x^4}} \over 4} + ....} \right)dx} $$ is equal to :

Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statements is/are TRUE about the function f(x) = max{3 $$-$$ x, x $$-$$ 1}?