Question

How do I find 2 unknown variables of f(x) when given the tangent line to a function at a point?

Find a and b if the tangent line to the function `f(x)=x^4`+a`x^2`+b at x=1 is y=2x+3

How do I find a and b in this equation?

Answer 1

Take the derivative of `f(x)` to find your slope at `x=1`, and solve for `a` then backsolve for `b`. you will find that `a=-1` and `b=5`

Explanation:

So we know by the definition of tangent that the tangent line should be equal to the function at `x=1`. We also know that the slope of the tangent line should be equal to the derivative of the function at the same point. Let's find the derivative of `f(x)` and plug in `x=1`:

`f(x)=x^4+ax^2+b`
`f'(x)=4x^3+2ax`

`f'(1)=4(1)^3+2a(1)`
`f'(1)=4+2a`

If the equation for the tangent line is `color(red)(y=2x+3`, then the slope of the line (which is equal to `f'(1)`) is 2.

`f'(1)=2=4+2a`

`-2=2a`

`color(blue)(a=-1)`

Now that we know `a`, we can go back and solve for `b`:

`f(1)=(1)^4+color(blue)((-1))(1)^2+b=color(red)(2(1)+3)`

`f(1)=1-1+b=color(red)(5)`

`color(green)(b=5)`