Find the maximum and minimum values, if any, of the following functions without using the derivatives :
(i) `f(x)=-(x-2)^(2)+3`
(ii) `f(x)=9x^(2)+12x+2.`

Answer

To find the maximum and minimum values of the given functions without using derivatives, we will analyze each function step by step.

### (i) Function: \( f(x) = -(x-2)^2 + 3 \)

1. **Identify the form of the function**: The function is in the form of a downward-opening parabola because of the negative sign in front of the squared term.

2. **Determine the vertex**: The vertex of a parabola in the form \( f(x) = a(x-h)^2 + k \) is given by the point \( (h, k) \). Here, \( h = 2 \) and \( k = 3 \). Thus, the vertex is at the point \( (2, 3) \).

3. **Maximum value**: Since the parabola opens downwards, the vertex represents the maximum point. Therefore, the maximum value of the function is \( f(2) = 3 \).

4. **Minimum value**: The function \( -(x-2)^2 \) can take any value less than or equal to 3, as \( (x-2)^2 \) is always non-negative. The minimum value of \( f(x) \) approaches negative infinity as \( x \) moves away from 2. Thus, there is no minimum value.

### Summary for (i):
- Maximum value: \( 3 \) at \( x = 2 \)
- Minimum value: None (approaches negative infinity)

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### (ii) Function: \( f(x) = 9x^2 + 12x + 2 \)

1. **Identify the form of the function**: This is a quadratic function in the standard form \( ax^2 + bx + c \), where \( a = 9 \), \( b = 12 \), and \( c = 2 \). Since \( a > 0 \), the parabola opens upwards.

2. **Calculate the discriminant**: The discriminant \( D \) is given by \( D = b^2 - 4ac \).
\[ D = 12^2 - 4 \cdot 9 \cdot 2 = 144 - 72 = 72 \]
Since \( D > 0 \), the quadratic has two distinct real roots.

3. **Find the vertex**: The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \).
\[ x = -\frac{12}{2 \cdot 9} = -\frac{12}{18} = -\frac{2}{3} \]

4. **Calculate the minimum value**: Substitute \( x = -\frac{2}{3} \) back into the function to find the minimum value.
\[ f\left(-\frac{2}{3}\right) = 9\left(-\frac{2}{3}\right)^2 + 12\left(-\frac{2}{3}\right) + 2 \]
\[ = 9 \cdot \frac{4}{9} - 8 + 2 = 4 - 8 + 2 = -2 \]

5. **Maximum value**: Since the parabola opens upwards, there is no maximum value; it approaches infinity as \( x \) moves away from the vertex.

### Summary for (ii):
- Minimum value: \( -2 \) at \( x = -\frac{2}{3} \)
- Maximum value: None (approaches infinity)

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