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Problem 76 Hard Difficulty

Suppose the curve $ y = x^4 + ax^3 + bx^2 + cx + d $ has a tangent line when $ x = 0 $ with equation $ y = 2x + 1 $ and tangent line when $ x = 1 $ with equation $ y = 2 - 3x. $ Find the values of $ a,b,c, $ and $ d. $

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he is clear. So when you read here, so we have why is equal to x square plus a X cubed plus B x square class C X plus t you have are given and we're gonna substitute X is equal to zero and why is equal to one? When we get into this equation and we get a d value of one were given that why is equal to two minus three x is attention Xs equal to one the y cornett when xs equal to one gives a Y value of negative one. So we see that one common negative one lies on the curb. So we're gonna substitute it into our main equation and we get negative too is equal to a plus B plus C plus de when we substitute t equals one. Yet negative three is equal to a plus B plus c Next we're going to difference. She our equation and we got d Y over DX is equal to four x cute plus three a x square plus to be explicit. See, it s stated that why equals two x plus one is a tangent at X equals two. So the slur slope of the curve at X equals two X equals zero is too. So we're gonna substitute two and we gotta see value of two and were given that like we stated before, why is equal to two minus three axes? A tangent at X equals one. So the slope of the curve X equals one is negative three. So we're also gonna substitute, do you? Why? Over D X is equal to negative three and X is equal to one. We get negative. Three is equal to four times one cubed plus three a one square plus to be terms one plus c. Then we get negative. C is equal to three a plus two three plus c When we substitute c last two and we get negative five is equal to a crust Be we're substituting it into the equation. Negative three plus a plus B plus c. So after we substitute to, we're gonna substitute to in our equation. Negatives. Seven equals three a plus two b plus C When we get negative. Nine as people 23 a plus two b When we get negative, 4.5 is equal to 1.5 a plus B and we're gonna substitute this with We're going to subtract thes two equations when we get you're me as one and we're going to substitute a equals one into negative five is equal to a plus B to get a B value of negative six. So after all this, we get our A, B, C and D values. And we're going to just grab this right here, just confirm our answer. Look, something like this guesses one common negative one and zero common one on our equation becomes why is equal to X to the fourth plus X Cube minus six square was two x plus one.