Answer to Question #145758 in Calculus for liam donohue

Question

The function
f(x)=−2x3+15x2+84x+4
is increasing on the interval (
,
).

It is decreasing on the interval ( −∞,
) and the interval (
, ∞ ).

The function has a local maximum at
.

Answer 1

f(x)=−2x³+15x²+84x+4

D(y): x є (−∞;∞)

f`(x)= -6x² +30x +84 = -x² +5x +14

Let's find the segments on which the function decreases and increases:

-x² + 5x + 14 > 0

Let us equate this equation to zero and find x. Find the roots of the equation using the Pythagorean theorem.

x₁ = -2

x₂ = 7

Let's draw a straight line and mark the points x1 and x2 on it, as well as substitute the numbers into these segments and calculate their values in the function.

Let's mark these points on the number line and calculate the value of the points in these intervals:

Total:

The function decreases in the interval:

(−∞;-2) U (7;∞)

And the function increases in the interval:

(-2;7)

In order to find a local maximum is necessary to substitute the values in which the derivative changes sign in the function f(x).

f(x)=−2x³+15x²+84x+4

f(7) > f(-2)

The function has a local maximum at x =7