Final answer:
The gradient of the curve y = √(x² - 10x + 26) will be 0 at a point where the derivative of y with respect to x is 0. After differentiating and setting the derivative equal to 0, we find that the slope is 0 at x = 5. Substituting x = 5 into the original function, we get y = 1, which does not match any of the provided options; hence, the correct answer is (D) None of the above.
Step-by-step explanation:
To find the coordinate of the point on the curve y = √(x² - 10x + 26) where the gradient is 0, we need to find the derivative of y with respect to x and then set it equal to 0. Since the derivative represents the gradient of the curve at any point, a gradient of 0 would indicate a horizontal tangent, hence a maximum or minimum point on the curve.
First, let's differentiate the function. Given the function is in the form of a square root, we can apply the chain rule for differentiation. If y = √(u), then dy/dx = (1/2√(u)) (du/dx). Here, u = x² - 10x + 26, so we need to find du/dx which is 2x - 10.
The derivative of y with respect to x (denoted as dy/dx) will be:
dy/dx = (1/(2√(x² - 10x + 26)))(2x - 10)
Setting the derivative equal to zero gives us:
(2x - 10)/(2√(x² - 10x + 26)) = 0
Since the denominator cannot be zero (as the square root of a real number is never negative), we only need to set the numerator equal to zero:
2x - 10 = 0
2x = 10
x = 5
Now, substitute x = 5 into the original equation to find y:
y = √(5² - 10· 5 + 26) = √(25 - 50 + 26) = √1 = 1
Unfortunately, none of the provided options match the point (5, 1). Therefore, the correct answer is (D) None of the above.