Find k such that the line is tangent to the graph of the function.

Function Line
f(x) = k√x y = 4x + 36

K= 1/2

To find the value of k such that the line is tangent to the graph of the function, we need to equate the slopes of the function and the line.

The slope of the line is given by the coefficient of x, which is 4.

The slope of the graph of the function can be found by taking the derivative of the function with respect to x.

So, let's find the derivative of f(x) = k√x with respect to x.

The derivative of f(x) = k√x can be found using the power rule of differentiation.

The power rule states that for any constant k and any positive exponent n, the derivative of kx^n is nkx^(n-1).

In this case, n = 1/2 and k is the constant.

So, differentiating f(x) = k√x with respect to x, we get:

f'(x) = k(1/2)x^(-1/2)

Now, let's set the slope of the function equal to the slope of the line:

k(1/2)x^(-1/2) = 4

To simplify the equation, let's square both sides to get rid of the fractional exponent:

[k(1/2)x^(-1/2)]^2 = 4^2

(k^2/4)x^(-1) = 16

Multiplying both sides by x to get rid of the x^(-1) term, we have:

k^2/4 = 16x

Now, let's solve for k:

k^2 = 64x

Taking the square root of both sides:

k = ±√(64x)

So, the value of k can be either √(64x) or -√(64x).

Now, we need to find the value of x at the point of tangency.

Since the line is tangent to the graph of the function, the x-coordinate of the point of tangency will be the value where the function and the line intersect.

Setting the equation for the function equal to the equation for the line, we have:

k√x = 4x + 36

Squaring both sides to get rid of the square root, we get:

k^2x = (4x + 36)^2

Now, substitute the value of k we found earlier: k = ±√(64x).

(k^2x) becomes (±√(64x))^2x = (±8√x)^2x = ±64x^2.

So, we have:

±64x^2 = (4x + 36)^2

Now, we can solve this equation for x using algebraic methods such as factoring, expanding and simplifying, or graphing.

Once you find the value(s) of x, you can substitute them back into the equation k = ±√(64x) to determine the value(s) of k.