Question

Show that the curve y = 4x³ + 7x - 8 has no tangent line with slope 3.

Answer 1

To show that the curve y = 4x³ + 7x - 8 has no tangent line with slope 3, we can use the derivative of the curve.

Step-by-step explanation:

To show that the curve y = 4x³ + 7x - 8 has no tangent line with slope 3, we can use the derivative of the curve. Taking the derivative of y with respect to x, we get y' = 12x² + 7. To find the slope of the tangent line at a particular point, we substitute the x-coordinate of that point into y'. If the slope is not equal to 3 for any x-value, it means there is no tangent line with slope 3.

Substituting the x-coordinate into y', we get 12x² + 7 = 3. Rearranging the equation, we have 12x² = -4. Dividing both sides by 12, we get x² = -1/3. Since x² cannot be negative, there are no real solutions for x. Therefore, there is no tangent line with slope 3 for the curve y = 4x³ + 7x - 8.

Answer 2

To demonstrate that the curve y = 4x³ + 7x - 8 has no tangent line with a slope of 3, we find its derivative (12x² + 7) and set it equal to 3. The resulting equation 12x² + 4 = 0 has no real solutions, indicating that there is no such tangent line with slope 3.

Step-by-step explanation:

To show that the curve y = 4x³ + 7x - 8 has no tangent line with a slope of 3, we need to find the derivative of y with respect to x. The derivative of a function gives us the slope of the tangent line at any point on the curve. Therefore, we need to take the derivative of the given function.

First, let's find the derivative: y' = d/dx (4x³ + 7x - 8) = 12x² + 7. This is because the derivative of x³ with respect to x is 3x², the derivative of x is 1, and the derivative of any constant is 0.

Next, we set the derivative equal to 3 to find any points where the tangent has a slope of 3: 12x² + 7 = 3. Simplifying this equation, we get 12x² + 4 = 0. This equation has no real solutions because a square term (x²) is always non-negative, and adding 4 to it cannot result in zero.

Since there are no real values of x that satisfy the equation, there is no point on the curve y = 4x³ + 7x - 8 where the tangent line has a slope of 3. Thus, we have shown that the curve has no tangent line with a slope of 3.