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