#### To determine

**To find:** The value of *c* if the given line is tangent to the curve.

#### Answer

The value of *c* = 6.

#### Explanation

**Given:**

The equation of line y=32x+6.

The curve is y=cx.

**Derivative rules:**

(1) Power Rule: ddx(xn)=nxn−1

(2) Constant multiple Rule: ddx[c⋅f]=cddx(f)

**Calculation:**

Obtain the slope of the curve.

The derivative of the curve y=cx is,

dydx=ddx(y)=ddx(cx)=ddx(cx12)

Apply the constant multiple rule (2) and the power rule (1),

dydx=cddx(x12)=c(12x12−1)=c(12x−12)=c2x

Thus, the derivative of the curve y=cx is c2x.

Therefore, the slope of the curve is c2x.

Obtain the value of *c*.

The slope of the given line y=32x+6 is 32.

Since the given line is tangent to the given curve, the slope of tangent to the curve at any point is equal to 32.

Every point of the curve is of the form (x,cx).

Since the point (a,ca) passes through both the tangent line and the curve, substitute the *a* for *x* in c2x=32,

c2a=32c=3(2a)2c=3a

Substitute (a,ca) for (x,y) in y=32x+6,

ca=32a+6

Substitute 3a for *c*,

(3a)a=32a+63a=32a+63a−32a=6(3−32)a=6

Simplify the terms,

(6−32)a=632a=6a=6×23a=4

Thus, the value of a=4.

Substitute 4 for *a* in c=3a,

c=34=3(2)=6

Therefore, the value of *c*= 6.