JEE Mains Maths · Application of Derivatives

Counting Real Roots

Using the derivative to count how many times a graph crosses the x-axis: a monotonic function crosses at most once, and otherwise the signs of the local maxima and minima decide.

All Application of Derivatives notes JEE Mains Maths strategy

Why this matters

Ten PYQs, half of them numerical answers. Four show that a function is monotonic, so it has at most one root; six find the local maximum and minimum values and count the sign changes between them. No equation here is solved — only counted. Two ideas cover the page.

Concept 1 of 2: A monotonic function has at most one root

f'>0

everywhere (or

f'<0

), the graph never turns, so it meets the axis at most once. To show it meets it once, find two points where

f

has opposite signs. A substitution such as

t=e^x

turns some equations into a polynomial on

t>0

; count only the roots in that range.

Definition

$f'$ of one sign: at most one real root.
$f(a)f(b)<0$ with $f$ continuous: a root in $(a,b)$ .
After $t=e^x$ , count roots with $t>0$ only.

At most one root

f'(x)>0\ \forall x\ \Rightarrow\ f(x)=0\ \text{has at most one root}

Worked example

How many real roots has

x^3+x+1=0

, and where?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 28 June 2022 · Q72Moderate

Example 1 · Application of Derivatives · Counting Real Roots

The number of real solutions of

x^{7}+ 5x^{3}+ 3x + 1 =

0 is equal to

At most one is not exactly one

Monotonicity gives at most one root. It is exactly one only when

f

also takes both signs — check two values, or the limits at the ends.

Drill 3 more on a monotonic function has at most one root

Concept 2 of 2: Counting roots from the extreme values

Between consecutive turning points the graph is monotonic, so it crosses the axis at most once there. List the local maxima and minima in order, with the limits at

\pm\infty

, and count the sign changes along the list. For

f(x)=k

, move the horizontal line

y=k

: a cubic has three real roots exactly when

k

lies strictly between its local minimum and maximum values.

Definition

Order: $f(-\infty)$ , the extreme values, $f(\infty)$ ; count sign changes.
Cubic $f(x)=k$ : three distinct roots when $f_{\min}<k<f_{\max}$ .
A local extreme value equal to 0 is a repeated root.

Three roots of a cubic

f(x)=k\ \text{has 3 distinct roots}\ \Leftarrow\ f_{\min}<k<f_{\max}

Worked example

For which

k

has

x^3-3x+k=0

three distinct real roots?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 14 June 2022 · Q161Moderate

Example 2 · Application of Derivatives · Counting Real Roots

The number of distinct real roots of the equation

x^{7}- 7x - 2 = 0

Count the ends too

The first and last crossings come from the limits at

\pm\infty

. An even-degree polynomial with positive leading term starts and ends positive, so a single minimum below 0 gives two roots, not one.

Drill 5 more on counting roots from the extreme values

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

A monotonic function has at most one root
At most one root
$f'(x)>0\ \forall x\ \Rightarrow\ f(x)=0\ \text{has at most one root}$
Counting roots from the extreme values
Three roots of a cubic
$f(x)=k\ \text{has 3 distinct roots}\ \Leftarrow\ f_{\min}<k<f_{\max}$

Watch out for (2)

At most one is not exactly one
→ A monotonic function has at most one root
Count the ends too
→ Counting roots from the extreme values

Test yourself on Application of Derivatives

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Application of Derivatives chapter test Drill all 10 questions