Let $$f(x) = {(\left| x \right| - \left| {x - 1} \right|)^2}$$

Which of the following equations is/are correct?

I. f($$-$$2) = f(5)

II. f''($$-$$2) + f''(0.5) + f''(3) = 4

Select the correct answer using the code given below.

Consider the function

$$f(x) = \left| {\matrix{ {{x^3}} & {\sin x} & {\cos x} \cr 6 & { - 1} & 0 \cr p & {{p^2}} & {{p^3}} \cr } } \right|$$,

where p is a contant.

What is the value of f'(0) ?

Consider the function

$$f(x) = \left| {\matrix{ {{x^3}} & {\sin x} & {\cos x} \cr 6 & { - 1} & 0 \cr p & {{p^2}} & {{p^3}} \cr } } \right|$$,

where p is a contant.

What is the value of p for which f''(0) = 0 ?

If $$y = {\log _{10}}x + {\log _x}10 + {\log _x}x + {\log _{10}}10$$, then what is $${\left( {{{dy} \over {dx}}} \right)_{x = 10}}$$ equal to ?