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Let h(x) = -3x⁴ + 8x³ + 18x²,
What is the absolute maximum value of h?

a. 270
b. 0
c. 135
d. h has no maximum value

Sagot :

Sqdancefanidn

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.

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