Manipal MET2010MathematicsApplication of Derivatives

The function f(x)= a x+b (x-1)(x-4) has a local maxima at (2,-1) , then

Options

Ab=1, a=0
Ba=1, b=0
Cb=-1, a=0
Da=-1, b=0

Correct answer

B. a=1, b=0

Step-by-step solution

Clearly, f(2)=-1 -1= 2 a+b (2-1)(2-4) 2 a+b=2 Now, gathered f^ (x)= 4 a+5 b-2 b x-a x^2 (x-1)^2(x-4)^2 f^ (2)=0 gathered b=0 a=1 f^ (x)=- (x-2)(x+2) (x-1)^2(x-4)^2 Clearly, for x 2, f^ (x) 0 and for x 2 , f^ (x) 0 . Thus, x=2 is indeed the point of local maxima for y=f(x)

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