(x, p) = a(x − p)^2 + b(x − p) + c, where a, b, c are constants and a < 0 and ‘p’ is a natural number. It is given that the roots of the equation ax^2 + bx +c = 0 are 3, 4. Then, the value of x at which f(x, 5) attains its maximum value is _______.Choices:- 9 9.5 7.5 5.5

The given equation is in the form of a quadratic equation. The maximum or minimum value of a quadratic equation ax^2 + bx + c is given by -b/2a.

Given that a < 0, the equation will have a maximum value.

The roots of the equation are given as 3 and 4. The sum of the roots of a quadratic equation is given by -b/a and the product of the roots is given by c/a.

So, we have:

3 + 4 = -b/a => -b/a = 7 => b = -7a

and

3*4 = c/a => c/a = 12 => c = 12a

Substituting these values in the equation for the maximum value, we get:

x = -b/2a = -(-7a)/2a = 7/2 = 3.5

But the question asks for the value of x at which f(x, 5) attains its maximum value. The function f(x, 5) is a shift of the original function f(x) by 5 units to the right. Therefore, the maximum value of f(x, 5) will occur at x = 3.5 + 5 = 8.5.

However, this is not one of the given options. There might be a mistake in the problem or in the given options.

The given equation is a quadratic equation in the form of f(x, p) = a(x − p)^2 + b(x − p) + c.

The maximum or minimum value of a quadratic equation ax^2 + bx + c is given by the vertex of the parabola, which is at x = -b/2a.

However, in this case, the equation is shifted by 'p' units, so the maximum value will be at x = p - b/2a.

Given that a < 0, the parabola opens downwards, so it will have a maximum value.

Also, it is given that the roots of the equation ax^2 + bx + c = 0 are 3 and 4. The sum of the roots of a quadratic equation is given by -b/a and the product of the roots is given by c/a.

So, we have -b/a = 3 + 4 = 7 and c/a = 3*4 = 12.

From -b/a = 7, we get b = -7a.

Substituting b = -7a in the equation for the maximum value, we get x = p - (-7a)/2a = p + 7/2.

Given that p = 5, we substitute p in the above equation to get x = 5 + 7/2 = 8.5.

So, the value of x at which f(x, 5) attains its maximum value is 8.5. However, this is not one of the given choices. There might be a mistake in the problem or the choices.