Show that the curve y = 4 x 3 + 7 x − 5 y=4x3+7x-5 has no tangent line with slope 2 2. y = 4 x 3 + 7 x − 5 ⇒ m = y ' = y=4x3+7x-5⇒m=y′= Preview , but x 2 x2 0 0 for all x x, so m ≥ m≥ for all x x.

The statement is true (see Step-by-step explanation).

Step-by-step explanation:

The slope of the tangent line for all point of the curve is determine by derive the expression abovementioned in the statement:

The previous expression represents a parabola, whose output will be positive for all due to the symmetry of and the positive coefficients of the polynomial. If the function is evaluated at , where the minimum occurs, it is evident that the smallest value is . Therefore, the inexistence of any tangent line with slope 2 associated with that curve is true.

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below:

y=1/4x-4

step-by-step explanation: