A graph has the general equation y= ax^3+bx+4.

Question

Two tangents that have the same gradient which is equal to 9.?

The first tangent touches the curve at x cordinate:-1, & the second tangent touched the curve at x cordinate: 3. Find the value of "a" and "b" in the curve function.

Answer 1

The curve y = ax³ + bx + 4

Differentiating  on each side with respect of x.

y' = 3ax² + b

Two tangents have same gradient  9.

So y' = 9

First tangent touches the curve at x = -1

Substitute x = -1 in y' = 3ax² + b.

y' = 3a(-1)² + b

y' = - 3a + b

9 = - 3a + b ---> (1)

Second tangent touches the curve at x = 3

Substitute x = 3 in y' = 3ax² + b.

y' = 3a(3)² + b

y' = 27a + b

9 = 27a + b ---> (2)

From (1) & (2)

- 3a + b = 27a + b

30a = 0

a = 0

Substitute a = 0 in equation (1).

9 = - 3(0) + b

b = 9

a = 0 and b = 9.