Answer:
c. 135
Step-by-step explanation:
You want the absolute maximum value of h(x) = -3x⁴ + 8x³ + 18x².
Maximum
The function will have a maximum where the derivative is zero.
h'(x) = -12x³ +24x² +36x
Factoring this gives ...
h'(x) = -12x(x² -2x -3) = -12x(x -3)(x +1)
The function h'(x) has zeros where its factors have zeros, at x=-1, x=0, x=3.
Shape
The graph of a quartic function with a negative leading coefficient is generally ∩-shaped. Because the derivative has three real zeros, we know the graph of h(x) is basically M-shaped, with 3 turning points.
The center zero (x=0) is closest to the left one (x=-1), which means the left peak of the graph is lower than the right peak. The maximum will be found where x=3.
h(3) = (3)²((-3·3 +8)(3) +18) = 9(-1(3) +18) = 135
The absolute maximum value of h(x) is 135, choice C.
