How to test the concavity of a composite squared function at a point

Key concept: When a function is built from another function, concavity comes from the second derivative, not just the graph shape. Here we differentiate carefully and then evaluate at the given point.

Step by step

We want to determine the concavity of

$y=(f(x))^2$

at $x=1$.

Step 1: Differentiate once

Use the chain rule:

$y' = 2f(x)f'(x)$

Step 2: Differentiate again

Apply the product rule to $2f(x)f'(x)$:

$y'' = 2\big((f'(x))^2 + f(x)f''(x)\big)$

Step 3: Evaluate at $x=1$

$y''(1)=2\big((f'(1))^2+f(1)f''(1)\big)$

Step 4: Decide concavity

• If $y''(1)>0$, the graph is concave up at $x=1$.
• If $y''(1)<0$, the graph is concave down at $x=1$.
• If $y''(1)=0$, the test is inconclusive.

So the answer depends on the sign of $2\big((f'(1))^2+f(1)f''(1)\big)$.

Pitfall alert

A common mistake is to stop after finding $y'=2f(x)f'(x)$. Concavity needs the second derivative. Another easy slip is forgetting that both $f(x)$ and $f'(x)$ change with $x$, so the product rule is required in the second differentiation.

Try different conditions

If the problem instead asked for concavity at a different point $x=a$, the same formula works:

$y''(a)=2\big((f'(a))^2+f(a)f''(a)\big)$

If you were only given $f(1)$ and $f'(1)$ but not $f''(1)$, then you would not have enough information to decide concavity.

Further reading

second derivative, chain rule, concavity