Theory and Examples

Determin...

Question

Theory and Examples

Determine the values of constants a and b so that f(x) = ax² + bx has an absolute maximum at the point (1,2).

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to find the values of M and N, so that the function H of X, which is equal to MX2 plus N X minus 5, has an absolute maximum at the point of -1.3. So we need to find the values of MNN so that we have an absolute maximum at the point of -1.3 for a function H of X. So the first thing we need to do is ensure that our function actually passes through the point of -1.3. That means that H of -1 has to be equal to 3. So therefore, we're going to have M multiplied by the quantity of minus 1 and quantity squared. Plus N multiplied by -1. Minus 5 has to be equal to 3. So simplifying this expression, we will have M minus N minus 5 equal to 3. We can add 5 to both sides of the equation, so we'll have M minus N is equal to 8. And now we can choose either variable to solve this equation for, so we'll solve this for M. That means that M is going to be equal to N + 8. Now, in order for us to have an absolute maximum at this point, it also needs to be a local maximum. That means our first derivative also needs to be zero at the point of X equals to minus 1. So, we need to calculate our first derivative, so we need to calculate each of X, and that's going to be equal to the derivative with respect to X. Of the quantity of M multiplied by X squared, plus N multiplied by X minus 5 in quantity. So, applying our constant multiple rule, as well as a concept rule for derivatives, this is going to be equal to 2 multiplied by M, multiplied by X plus N. So now we need to set this first derivative equal to 0, when X is equal to -1. So, H of -1, which is going to be equal to. 2 multiplied by M, multiplied by minus 1. Plus in And that's going to be equal to. N minus 2 M, and this quantity has to be equal to 0, so we'll set N minus 2 M equal to 0. And here we can go ahead and substitute in the expression for M. so we'll have N. Minus 2 multiplied by the quantity of N + 8 in quantity, that's gonna be equal to 0. Distributing the minus 2 will have N minus 2N minus 16 is equal to 0. Combining like terms, we'll have minus n, minus 16 equal to 0, and the sulfurn we'll just add N to both sides, and then we get N as equal to -16. So that is one of our variables found. And so now we can take this value of N and plug it back into our expression for M. So that means that M is equal to -16 + 8, and so M is equal to -8. So those are the two values that we're looking for M equal to -8 and N is equal to -16. And this does indeed give us an absolute maximum. Since the coefficient of the quadrag term is negative, our M value is negative, that means that our parabola is opening up downwards. So that means that the local maximum that we found is going to be an absolute maximum. So I hope that this explanation has been useful, and I'll see you in the next video.

Theory and Examples

Determine the values of constants a and b so that f(x) = ax² + bx has an absolute maximum at the point (1,2).

Key Concepts

Absolute Maximum

Derivative and Critical Points

Substitution and System of Equations

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