Prove that the graph of a quadratic polynomial function f(x)=a x^2+b x+c, a ≠0, is concave upwar

Prove that the graph of a quadratic polynomial function...

Key Concepts

Second Derivative Test

The second derivative test is a method used in calculus to determine the nature of critical points by examining the second derivative at those points. For quadratic functions, since the second derivative is constant, its sign (determined by a in 2a) dictates the overall concavity of the graph.

Second Derivative

The second derivative of a function is the derivative of its first derivative and indicates the concavity of the function's graph. For a quadratic function f(x) = ax² + bx + c, the second derivative is constant and equals 2a, which simplifies the analysis of the function's curvature.

Concavity

Concavity describes the direction in which a function curves. If a function is concave upward, its graph forms a 'U' shape, and if it is concave downward, its graph forms an 'n' shape. The sign of the second derivative is used to determine whether the function is concave upward (positive second derivative) or concave downward (negative second derivative).

Quadratic Functions

A quadratic function is any function in the form f(x) = ax² + bx + c, where a, b, and c are constants with a ? 0. The coefficient a plays a crucial role in determining the direction and width of the parabola formed by the graph of the quadratic function.

Derivatives

The derivative of a function represents its rate of change and is a fundamental concept in calculus. For quadratic functions, computing the first derivative shows how the function's slope changes at any given point, while the second derivative provides insight into the curvature of the function.

Transcript

00:01 Let's look at the second order polynomial or quadratic function, that a x squared plus bx plus c.

00:08 And they ask us about the curvature of this.

00:11 And so to take two derivatives, we take the first derivative is 2a times x plus b, and then a second derivative is 2a.

00:17 And so they ask us to show that, well, they established that if a is negative, then this is concave down.

00:27 And in fact, we can see if a is negative, then f double prime is negative.

00:31 So we have concave down.

00:33 And if a is positive, then f double prime is positive, and we have concave up.

00:37 So the sign, sgn of f double prime, is the same as the sign of a.

00:42 So if a is positive, we have a problem that goes up.

00:46 And it's just, you know, a and b and c, and i mean, these just kind of shift, shift how steep it is and shift around where it is.

00:54 And if a is negative, then it's always going to go down.

00:59 Now, they ask us about, let's see, that be a function for which the third derivative exists on an interval a to b that contains the number c.

01:09 So c is inside of a to b and it's continuous and the third derivative exists in that range.

01:14 And we're told that fc double prime is zero.

01:17 So there is a possible inflection point there.

01:22 Now let's see here.

01:25 But we're told that fc triple prime is not equal to zero.

01:30 Well what that says is that the triple prime is not equal to zero and double prime equals zero.

01:35 Then that means double prime has to change sign as it goes from less than c to greater than c.

01:43 Because it has a, if you just think of this as the function, right, then this is the slope of that function.

01:50 And if the slope is not zero and the function is zero, then it has to change sign.

01:56 So you can just look at this as like just another function g, you know, f double prime is, is equal to f double prime and take a derivative of that, that's f triple prime.

02:06 So if f double prime changes sign, then that means that we have an inflection point because we have going from one type of from negative to positive or positive to negative...

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