Q. 93 - Calculus

Question

Prove that every quadratic function is either always concave up or always concave down.

Step-by-Step Solution

Answer

Proved that every quadratic function is either always concave up or always concave down. 

Step 1. Given Information.

Given a quadratic function. Let that function be $f\left(x\right) = a{x}^2+bx+c.$

Step 2. Theorem.

The Derivative Measures Where a Function is Increasing or Decreasing

Let f be a function that is differentiable on an interval I.

(a) If $f\text{'}$ is positive in the interior of I, then f is increasing on I.

(b) If $f\text{'}$ is negative in the interior of I, then f is decreasing on I.

(c) If $f\text{'}$ is zero in the interior of I, then f is constant on I.

Step 3. Proof.

$$\begin{gathered} f\left(x\right) = a{x}^2+bx+c, \\ Differentiating both sides w.r.t x we get, \\ f\text{'}\left(x\right) = 2ax+b, \\ Differentiating both sides w.r.t x once again we get, \\ f\text{'}\text{'}\left(x\right) = 2a, \\ Now, f\text{'}\text{'}\left(x\right) is a cons\tan t and does not depend on x and according to the \\ value of cons\tan t a it will be either positive or negative and accordingly the \\ function f\left(x\right) will be concave upward or concave downward respectively. \end{gathered}$$

Hence, Proved.