JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

If the functions $f(x) = \frac{x^2}{3} + 2bx + \frac{ax^2}{2}$ and $g(x) = \frac{x^2}{3} + ax + bx^2$ , where $a \neq 2b$ , have a common extreme point, then the value of $a + 2b + 7$ is equal to:

Choose the correct answer:

Explanation

olution

1. Differentiate the functions:

$f'(x) = x^2 + ax + 2b$
$g'(x) = x^2 + 2bx + a$

2. Set both to zero for a common extreme point $x_0$ :

Since both equal zero, we can set them equal to each other:

x_0^2 + ax_0 + 2b = x_0^2 + 2bx_0 + a

3. Solve for $x_0$ :

Subtract $x_0^2$ from both sides:

ax_0 + 2b = 2bx_0 + a

ax_0 - 2bx_0 = a - 2b

x_0(a - 2b) = (a - 2b)

Since $a \neq 2b$ , we can divide both sides by $(a - 2b)$ :

x_0 = 1

4. Substitute $x_0 = 1$ back into either derivative:

Using $f'(1) = 0$ :

1^2 + a(1) + 2b = 0

1 + a + 2b = 0

a + 2b = -1

5. Calculate the final value:

The question asks for $a + 2b + 7$ :

-1 + 7 = 6

Choose the correct answer:

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