Q.Find the points at which the function $f$ given by $f(x) = (x - 2)^4 (x + 1)^3$ has

Idea: Differentiate, factor, and use the sign change of $f'$ (first derivative test) at each critical point.

$f(x)=(x-2)^4(x+1)^3$. By the product rule,

Factor out $(x-2)^3(x+1)^2$:

Critical points: $x=-1,\ x=\tfrac{2}{7},\ x=2$.

Sign of $f'$ (note $(x+1)^2\ge0$ never changes sign):

• $x<-1$: $(-)(+)(-)=+$
• $-1<x<\tfrac27$: $(-)(+)(-)=+$
• $\tfrac27<x<2$: $(-)(+)(+)=-$
• $x>2$: $(+)(+)(+)=+$

Classify: …

The plan

To locate maxima, minima and inflexions we look at where the slope $f'(x)$ is zero and, crucially, how the sign of $f'$ changes there. Positive-to-negative means a peak (local max); negative-to-positive means a valley (local min); no change means a horizontal point of inflexion.

Step 1 — Differentiate and factor

Using the product rule,

Both terms share $(x-2)^3(x+1)^2$, so

Simplify the bracket: $4x+4+3x-6=7x-2$. Hence

Step 2 — Critical points

Set $f'(x)=0$:

In increasing order these are $x=-1,\ \tfrac27,\ 2$.

Step 3 — Sign chart of $f'$

The factor $(x+1)^2$ is never negative, so it cannot switch the sign of $f'$ — it only makes $f'$ vanish at $x=-1$. The sign of $f'$ is therefore controlled by $(x-2)^3$ (same sign as $x-2$) and $(7x-2)$.

Step 4 — Classify each critical point …

Method: Classifying Critical Points Using Root Multiplicity in the First Derivative

This method locates and classifies every local maximum, local minimum, and point of inflexion of a function whose derivative factors into repeated linear factors — very common when the original function is itself a product of powers, like $(x-a)^m(x-b)^n$.

Steps

Step 1: Differentiate using the product rule and factor out the common part

For $f(x)=(x-p)^m(x-q)^n$, apply the product rule and notice both resulting terms share the factor $(x-p)^{m-1}(x-q)^{n-1}$ — pull it out to leave a simple linear bracket behind.

Step 2: Identify every critical point, keeping track of each factor's multiplicity (exponent)

Set each factor of $f'(x)$ to zero. Note the exponent on each repeated factor — this exponent is the key to classification in the next step.

Step 3: Build a sign chart, using the multiplicity rule to shortcut the sign of each factor …

Common Mistakes

Mistake 1: Assuming every zero of $f'(x)$ is automatically a local maximum or minimum

Why it's wrong: a root of $f'$ coming from a factor raised to an even power (like $(x+1)^2$ here) makes $f'=0$ at that point but does NOT change the sign of $f'$ on either side, so the curve keeps rising (or keeps falling) through it — it's a point of inflexion, not an extremum. Correct approach: always check the sign of $f'$ on both sides of every critical point before classifying it; the exponent's parity is a fast shortcut, but only once you've confirmed why it works with the sign chart.

Mistake 2: Making an arithmetic slip when combining the two product-rule terms into one bracket …