The cubic function's graph has a relative minimum at (0, 6), a relative maximum at (7/6, 49/27), and exhibits increasing and decreasing behavior in specific intervals. The end behavior indicates that f(x) approaches +infinity as x approaches +infinity and f(x) approaches -infinity as x approaches -infinity.
To graph the function f(x) = 2x^3 - 7x^2 + 6, let's analyze its key characteristics and use a graphing calculator for precision.
Domain and Range:
The domain is the set of all real numbers since there are no restrictions on the input values for a cubic function. The range may include all real numbers, as cubic functions generally cover the entire range.
Relative Maximum(s) and Minimum(s):
To find turning points, we take the derivative and set it equal to zero. By solving f'(x) = 6x^2 - 14x, we get critical points at x = 0 and x = 7/6. Substituting these into the original function gives f(0) = 6 and f(7/6) = 49/27. So, the relative minimum is (0, 6), and the relative maximum is (7/6, 49/27).
End Behavior:
As x approaches positive or negative infinity, the function behaves like a cubic polynomial. For large positive x, f(x) approaches +infinity, and for large negative x, f(x) approaches -infinity.
Increasing and Decreasing Intervals:
By examining the sign of f'(x), we find that f(x) is increasing on (-infinity, 0) and decreasing on (0, 7/6), and increasing again on (7/6, +infinity).
In summary, the function f(x) = 2x^3 - 7x^2 + 6 has a relative minimum at (0, 6), a relative maximum at (7/6, 49/27), and specific intervals of increasing and decreasing behavior. The end behavior suggests that f(x) approaches +infinity as x goes to positive infinity and f(x) approaches -infinity as x goes to negative infinity.