To determine the value of
λ such that
f(x)=2x3+9x2+λx+20 is a decreasing function in the largest possible interval
(−2,−1), we need to consider the first derivative of the function.
The first derivative of
f(x) is given by:
For the function to be decreasing in the interval
(−2,−1), the first derivative
f′(x) must be less than or equal to zero in that interval. Therefore, we need:
f′(x)≤0‌ for ‌x∈(−2,−1)This gives us the inequality:
6x2+18x+λ≤0‌ for ‌x∈(−2,−1) To find the value of
λ such that the inequality is satisfied, we need to test the boundaries of the interval
(−2,−1). At the endpoints:
When
x=−2 :
f′(−2)=6(−2)2+18(−2)+λ=24−36+λ=λ−12
For the point
x=−2, we want:
λ−12≤0⟹λ≤12When
x=−1 :
f′(−1)=6(−1)2+18(−1)+λ=6−18+λ=λ−12
For the point
x=−1, we want:
λ−12≤0⟹λ≤12 Since we need the inequality to hold throughout the interval and we have obtained the same condition from both endpoints,
λ must be 12 to ensure the function is decreasing throughout the interval
(−2,−1).
Therefore, the correct answer is:
Option C
λ=12