work out the coordinates of the turning point of the curve y=7x-x^2-5.  give any fractions in their simplest forms

The equation of the curve is \( y = 7x - x^2 - 5 \).

To find the turning point, we first differentiate the equation to obtain the gradient function and then solve for \( x \) when the gradient is zero (since the turning point occurs when the gradient is zero).

1. **Differentiate \( y = 7x - x^2 - 5 \)** with respect to \( x \):
   \[
   \frac{dy}{dx} = 7 - 2x
   \]

2. **Set the derivative equal to zero** to find the \( x \)-coordinate of the turning point:
   \[
   7 - 2x = 0
   \]
   \[
   2x = 7
   \]
   \[
   x = \frac{7}{2}
   \]

3. **Substitute \( x = \frac{7}{2} \) back into the original equation** to find the corresponding \( y \)-coordinate:
   \[
   y = 7\left(\frac{7}{2}\right) - \left(\frac{7}{2}\right)^2 - 5
   \]
   \[
   y = \frac{49}{2} - \frac{49}{4} - 5
   \]
   To simplify, express all terms with a common denominator of 4:
   \[
   y = \frac{98}{4} - \frac{49}{4} - \frac{20}{4}
   \]
   \[
   y = \frac{98 - 49 - 20}{4}
   \]
   \[
   y = \frac{29}{4}
   \]

Thus, the coordinates of the turning point are:
\[
\left( \frac{7}{2}, \frac{29}{4} \right)
\]

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How do you determine whether the turning point is a maximum or minimum?
2. Can you sketch the graph of this quadratic equation?
3. What are the roots of the equation \( 7x - x^2 - 5 = 0 \)?
4. How does completing the square help in finding the turning point of a quadratic function?
5. How would the turning point change if the equation were \( y = 7x - 2x^2 - 5 \)?

**Tip**: The second derivative of a function helps determine whether a turning point is a maximum or a minimum.

Work out the coordinates of the turning point of the curve y = 7x - x^2 - 5. Give any fractions in their simplest forms.

Discover how to calculate the turning point of the quadratic curve y = 7x - x^2 - 5. This solution involves key calculus concepts like differentiation and the first derivative test, ideal for students in Grades 10-12. Learn to find the exact coordinates using fractions in simplest form.

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