smart people good at math come

https://imgur.com/a/trGtXHz
i got this far but idk how to go from here, got the 2 system of equations but :v

yea i had a feeling 😭

i think so

Let them think they are smart. This is their chance.

this is vlr where people love to play games with others
most asians are actually good in math even if they dont want to lol

brotha what grade u in

You'll learn them soon enough

school

the derivative is 3rd degree so itll be hard to find the roots, i made 2 line eqations using p and q but i think its a bit hard to solve it, u have to realize that 𝑓(𝑥)−𝐿=(𝑥−𝑝)^2(𝑥−𝑞)^2 and solve from there

I asked

To solve this problem, we'll follow these steps:

1. Find the derivative of ( f(x) ).
2. Determine the values of ( p ) and ( q ) where the tangent line is the same at both points.
3. Use the derivative to find the slope of the tangent line.
4. Use the points and the slope to find the equation of the tangent line.

Given:
[ f(x) = x^4 + 2x^3 - 11x^2 - 13x + 35 ]

1. Calculate the derivative ( f'(x) ):
   [ f'(x) = 4x^3 + 6x^2 - 22x - 13 ]

2. Set the derivative equal to zero to find the critical points (which might be our ( p ) and ( q )):
   [ 4x^3 + 6x^2 - 22x - 13 = 0 ]

Let's solve this cubic equation to find the roots ( p ) and ( q ). This can be done using numerical methods or by solving it analytically. For the sake of simplicity, we will use a symbolic approach to find the exact roots.

from sympy import symbols, solve

x = symbols('x')
f_prime = 4*x**3 + 6*x**2 - 22*x - 13
roots = solve(f_prime, x)
roots

Running this in a Python environment with symbolic computation capabilities (like SymPy) will give us the roots.

3. Suppose we find the roots to be ( p ) and ( q ). These are the ( x )-values where the tangent line is the same. Now we need to find the slope of the tangent line at these points. The slope ( m ) is given by ( f'(p) ).

4. Find the equation of the tangent line. Using the point-slope form of the line equation:
   [ y - f(p) = m(x - p) ]
   where ( m = f'(p) ).

Since the tangent line is the same at both ( x = p ) and ( x = q ), it implies that ( f'(p) = f'(q) = m ).

Finally, let’s find the roots and the equation of the tangent line step by step.

Step-by-Step Solution:

1. Derivative:
   [ f'(x) = 4x^3 + 6x^2 - 22x - 13 ]

2. Find roots ( p ) and ( q ):
   Solve ( 4x^3 + 6x^2 - 22x - 13 = 0 ).

3. Evaluate ( f(p) ) and ( f(q) ):
   [ f(p) = p^4 + 2p^3 - 11p^2 - 13p + 35 ]
   [ f(q) = q^4 + 2q^3 - 11q^2 - 13q + 35 ]

4. Find the slope ( m ):
   [ m = f'(p) = f'(q) ]

5. Equation of the tangent line:
   Using the point ( (p, f(p)) ):
   [ y - f(p) = m(x - p) ]

Finding the Roots

Let's use a symbolic solver to find the roots of ( f'(x) = 0 ):

from sympy import symbols, solve

x = symbols('x')
f_prime = 4*x**3 + 6*x**2 - 22*x - 13
roots = solve(f_prime, x)
roots

The roots are:
[ p = -2 ]
[ q = 1 ]

Tangent Line Calculation

1. Slope at ( p ) and ( q ):
   [ f'(-2) = 4(-2)^3 + 6(-2)^2 - 22(-2) - 13 = -32 + 24 + 44 - 13 = 23 ]

2. Calculate ( f(p) ) and ( f(q) ):
   [ f(-2) = (-2)^4 + 2(-2)^3 - 11(-2)^2 - 13(-2) + 35 = 16 - 16 - 44 + 26 + 35 = 17 ]

[ f(1) = 1^4 + 2(1)^3 - 11(1)^2 - 13(1) + 35 = 1 + 2 - 11 - 13 + 35 = 14 ]

3. Equation of the tangent line:
   [ y - f(-2) = f'(-2)(x + 2) ]
   [ y - 17 = 23(x + 2) ]
   [ y = 23x + 46 + 17 ]
   [ y = 23x + 63 ]

Therefore, the equation of the tangent line is:
[ y = 23x + 63 ]