Concept:

The points where the f'(x) = 0 are known as critical values.

At critical point, if f ''(x) > 0 then function has minima.

If f ''(x) < 0 then function has maxima.

Calculation:

For x ≥ 0

f(x) = − 3sin x

⇒ f '(x) = − 3cos x

⇒ f ′(0) = − 3

For x < 0, f (x) = x³ + x² + 10x

⇒ f ′(x) = 3x² + 2x + 10

⇒ f ′(0) = 10

⇒ f ′(x) > 0 for x < 0 and f ′(x) < 0 for x ≥ 0

⇒ f (x) has maxima at x = 0.

∴ At x = 0, there is a point of maximum.

The correct answer is Option 1.

Explanation:

1. Given: y² = 3x --- (1)

and x² = 3y --- (2)

∴ x = 0 , 3 

At x = 0,  y = 0 and at x = 3, y =3

Therefore, the points of intersection of the given curves are (0, 0) and (3, 3) 

∴ Statement 1 is correct.

Concept:

Order of a Differential Equation: The order is determined by the highest derivative present in the equation.

Degree of a Differential Equation: The degree is the power of the highest-order derivative after it has been made free of any fractional or radical exponents.

Calculation:

Given differential equation

The order of this differential equation will be 3. Also, the degree of this will be 1 

∴ Option 1 is correct

Solution:

Statement I: Greatest integer function is differentiable on R except for integer points.

The greatest integer function is not continuous at the integers level and any function which is discontinuous at the integer value, will be non-differentiable at the point.

As the value jumps at each integral value, therefore, it is discontinuous at each integral value.

Hence, Statement I is correct.

Statement II: Fractional part function is differentiable on R except for integer points.

It is not possible to differentiate the fractional part of x when x ∈ Z.

This is because the graph of {x} is not continuous. So its derivative does not exist.

If we look at the right-hand derivate and left-hand derivate of the integral values of x, they are not the same.

For a function to be differentiated, the left-hand derivate and right-hand derivate must be the same.

Hence, its derivative does not exist at x ∈ Z.

However, other than the integral values of {x} the derivative exists.

∴ Statement II is correct

So, the correct option is (3)

Concept:

The equation of tangent of ax₂ + by₂ + 2hx +2gy + c = 0 at (x₁, y₁) is axx₁ + byy₁ + 2h 

Calculation:

Given, x + √3y = 2√3 is the tangent at the point (3√3/2, 1/2)

⇒ x₁ = 3√3/2 and y₁ = 1/2

⇒ x(3√3/2) + √3y(3√3/2) = 2√3(3√3/2)

⇒ x(3√3/2) + 9y(1/2) = 9

⇒ xx₁ + 9yy₁ = 9, which is tangent to x² + 9y² = 9

∴ The line x + √3y = 2√3 is the tangent at the point (3√3/2, 1/2) for x² + 9y² = 9.

The correct answer is Option 1.