Functions & Graphs (Mixed problems)

Let f(x) = 121 – x², g(x) = |x – 8| + |x + 8| and h(x) = min {f(x), g(x)}. What is the number of integer values of x for which h(x) is equal to a positive integral value?

If the function R(x) = max (x² – 8, 3x, 8), then what is the max value of R(x)?

If the function R(x) = min (x² – 8, 3x, 8), what is the max value of R(x)?

The minimum value of ax² + bx + c is 7/8 at x = 5/4. Find the value of the expression at x = 5, if the value of the expression at x = 1 is 1.

The function f(x) is defined for positive integers and is defined as:

f(x) = 6^x – 3, if x is a number in the form 2n.

       = 6^x + 4, if x is a number in the form 2n + 1.

What is the remainder when f(1) + f(2) + f(3) + … + f(1001) is divided by 2?

p, q and r are three non-negative integers such that p + q + r = 10. The maximum value of pq + qr + pr + pqr is

For a positive integer x, f(x + 2) = 3 + f(x), when x is even and f(x + 2) = x + f(x), when x is odd. If f(1) = 6 and f(2) = 4, then find (f(f(f(f(1)))) × f(f(f(f(2)))).

If x > 0, the minimum value of

\( \dfrac{\left(x+\dfrac{1}{x}\right)^6-\left(x^6+\dfrac{1}{x^6}\right)-2}{\left(x+\dfrac{1}{x}\right)^3+\left(x^3+\dfrac{1}{x^3}\right)} \)        is ____________ ?

If [x] denotes the greatest integer ≤ x, then

If x and y are real numbers, then the minimum value of x² + 4xy + 6y² – 4y + 4 is