To find the range of values of X for which the function is increasing, we need to find the derivative of the function and determine when the derivative is positive.
The derivative of the function Y = 8X + X^2 - (X^3/3) is:
Y' = 8 + 2X - X^2
To find when the derivative is positive, we set Y' > 0 and solve for X:
8 + 2X - X^2 > 0
Rearranging the inequality:
X^2 - 2X - 8 < 0
Factoring, we get:
(X - 4)(X + 2) < 0
This inequality represents when the function is increasing. To solve for X, we set each factor equal to 0 and find the sign changes:
X - 4 = 0, X = 4 (sign change from negative to positive)
X + 2 = 0, X = -2 (sign change from positive to negative)
Plotting these points on a number line:
-∞ -2 4 ∞
Since the inequality is less than 0, the function is increasing when X is between -2 and 4.
Therefore, the range of values of X for which the function is increasing is -2 < X < 4.
Find the range of values of X for which the function is increasing
Y=8X+X^2-X^3/3
1 answer
1 answer