To find the coordinates of the turning point on the curve given by the equation \(\( y = -4x^2 + 4x + 5 \)\), we need to follow a series of steps involving differentiation and solving equations. Here's the step-by-step solution:
### Step 1: Find the first derivative \(\( \frac{dy}{dx} \)\)
First, we need to differentiate \(\( y \)\) with respect to \(\( x \)\):
\(\[ y = -4x^2 + 4x + 5 \]\)
The power rule for differentiation \(\( \frac{d}{dx} [ax^n] = n \cdot ax^{n-1} \)\) will be applied to each term:
\(\[ \frac{dy}{dx} = \frac{d}{dx}(-4x^2) + \frac{d}{dx}(4x) + \frac{d}{dx}(5) \]\)
\(\[ \frac{dy}{dx} = -8x + 4 \]\)
### Step 2: Set the first derivative equal to zero
To find the turning points, we need to set the first derivative equal to zero and solve for \(\( x \)\):
\(\[ -8x + 4 = 0 \]\)
Rearranging to solve for \(\( x \)\):
\(\[ -8x + 4 = 0 \]\)
\(\[ -8x = -4 \]\)
\(\[ x = \frac{1}{2} \]\)
### Step 3: Substitute \(\( x = \frac{1}{2} \)\) back into the original equation to find \(\( y \)\)
Next, we substitute \(\( x = \frac{1}{2} \)\) back into the original equation \(\( y = -4x^2 + 4x + 5 \)\) to find the corresponding \(\( y \)\):
\(\[ y = -4\left(\frac{1}{2}\right)^2 + 4\left(\frac{1}{2}\right) + 5 \]\)
\(\[ y = -4 \cdot \frac{1}{4} + 4 \cdot \frac{1}{2} + 5 \]\)
\(\[ y = -1 + 2 + 5 \]\)
\(\[ y = 6 \]\)
### Step 4: Write down the coordinates of the turning point
Thus, the coordinates of the turning point are:
\(\[ \left(\frac{1}{2}, 6\right) \]\)
### Summary
By differentiating the given function, setting the derivative equal to zero, and solving for \(\( x \)\), we found that the turning point occurs at \(\( x = \frac{1}{2} \)\). Substituting \(\( x = \frac{1}{2} \)\) back into the original function gives us the corresponding \(\( y \)\)-value of 6. Hence, the coordinates of the turning point are:
\(\[ \left(\frac{1}{2}, 6\right) \]\)
This completes our detailed, step-by-step solution.
