How to check whether a product function has a horizontal tangent at a point

Key concept: For a product like $f(x)g(x)$, a horizontal tangent happens exactly when the derivative is zero at that $x$-value. The product rule is the key move.

Step by step

To determine whether $y=f(x)g(x)$ has a horizontal tangent at $x=1$, use the derivative.

Step 1: Differentiate with the product rule

$y' = f'(x)g(x) + f(x)g'(x)$

Step 2: Evaluate at $x=1$

$y'(1)=f'(1)g(1)+f(1)g'(1)$

Step 3: Check the result

• If $y'(1)=0$, then the graph has a horizontal tangent at $x=1$.
• If $y'(1)\neq 0$, then it does not.

So the answer depends on the values of $f(1)$, $g(1)$, $f'(1)$, and $g'(1)$. Without those numbers, you cannot decide yes or no.

Pitfall alert

Don’t assume a product has a horizontal tangent just because one factor is zero. You need the derivative at the point, not just the function value. Also, if the problem gives only the formula and no values, the question is incomplete.

Try different conditions

If you were told that $f(1)=2$, $f'(1)=0$, $g(1)=-3$, and $g'(1)=4$, then

$y'(1)=0\cdot(-3)+2\cdot 4=8$

so there would be no horizontal tangent. If instead the derivative came out to $0$, then the tangent would be horizontal.

Further reading

product rule, horizontal tangent, derivative at a point