If y=a x+b x²+x has its extremum values at x=-1 and x=2, then

Correct Answer is : a=2,b=−12a=2, b=-1/2a=2,b=−21

Correct Answer is : a=2,b=−12a=2, b=-1/2a=2,b=−21

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MHTCET 16 May 2024 Shift 1 Paper

Question : 118 of 150

Marks: +1, -0

y=a \log x+b x^{2}+x

x=-1

x=2

, then

Solution:

Concept:

For extremum values, the first derivative

\frac{dy}{dx}

must be zero at the given points

x=-1

and

x=2

Explanation:

Given

y = a\log x + bx^2 + x

Differentiate with respect to

x

\frac{dy}{dx} = \frac{a}{x} + 2bx + 1.

Since

x=-1

is an extremum point,

\frac{dy}{dx}=0

x=-1

-a - 2b + 1 = 0 \Rightarrow a + 2b = 1.

Since

x=2

is an extremum point,

\frac{dy}{dx}=0

x=2

\frac{a}{2} + 4b + 1 = 0 \Rightarrow a + 8b + 2 = 0.

Subtract the first equation from the second:

(a+8b+2) - (a+2b) = 0 - 1 \Rightarrow 6b = -3 \Rightarrow b = -\frac{1}{2}.

Substitute

b = -\frac{1}{2}

into

a+2b=1

a - 1 = 1 \Rightarrow a = 2.

Thus,

a = 2

and

b = -\frac{1}{2}

Answer:

a = 2, b = -\frac{1}{2}

Therefore, the correct option is B.

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