The set of all real values of a for which the function f(x)=(a+2)x3−3ax2+9ax−1 decreases monotonically throughout for all real x, is
A
a<−2
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B
a>−2
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C
−3≤a<0
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D
a≤−3
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Solution
The correct option is Da≤−3 We have f(x)=(a+2)x3−3ax2+9ax−1 ⇒f′(x)=3(a+2)x2−6ax+9a=3[(a+2)x2−2ax+3a] Now, f′(x)≤0∀x∈R ⇒a<−2 and discriminant ≤0 ⇒4a2≤12a(a+2) ⇒a2≤3a2+6a ⇒2a2+6a≥0 ⇒2a(a+3)≥0 ⇒a∈(−∞,−3]∪[0,∞) But a<−2. Hence a∈(−∞,−3]