MYTUTOR SUBJECT ANSWERS

285 views

Given that a and b are distinct positive numbers, find a polynomial P(x) such that the derivative of f(x) = P(x)e^(−x²) is zero for x = 0, x = ±a and x = ±b, but for no other values of x.

This is a question from a STEP II paper.

STEP questions always give you just enough information to solve the problem; we’re looking to use everything that we’re given. The first thing we should do is differentiate f(x) by the product rule. This is a step up from A-level since we’re considering a general function P(x), but the product rule still works just as usual:

f’(x) = P’(x)e^(-x²) - 2xP(x)e^(−x²) = e^(−x²)[P’(x) – 2xP(x)]

Now we can set the derivative we’ve found equal to zero for x = 0, ±a and ±b.

       •   f’(0) = P’(0) = 0

       •   f’(a) = e^(-a²)[P’(a) – 2aP(a)] = 0  ⇒  P’(a) - 2aP(a) = 0

       •   f’(-a) = e^(-a²) [P’(-a) + 2aP(-a)] = 0  ⇒  P’(-a) + 2aP(-a) = 0

       •   f’(b) = e^(-b²)[P’(b)– 2bP(b)] = 0  ⇒  P’(b) - 2bP(b) = 0

       •   f’(-b) = e^(-b²)[P’(-b) + 2bP(-b)] = 0  ⇒  P’(-b) + 2bP(-b) = 0

The only information we’re given that’s left is that the derivative of f(x) isn’t zero for any other values of x. So e^(−x²)[P’(x) – 2xP(x)] ≠ 0 for any other values of x. Since e^(−x²) is always non-zero we can deduce that P’(x) – 2xP(x) ≠ 0 for any other values of x.

We can now combine everything we know together: P’(x) - 2xP(x) = 0 ONLY for x = 0, ±a, ±b.

How do we proceed now? We’ve used all the information given in the question. Let’s look back at what we’re asked to do: we’re asked to find a polynomial P(x) which satisfies the above conditions. The trick here is to equate the polynomial P’(x) – 2xP(x) to a polynomial that we already know equals zero ONLY for x = 0, ±a, ±b. Then by comparing coefficients we can find coefficients for P(x). Let’s use x(x - a)(x + a)(x - b)(x + b) = x(x² - a²)(x² - b²) = x⁵ - (a² + b²)x³ + a²b²x.

What order does P(x) need to be? The order of x⁵ - (a² + b²)x³ + a²b²x is 5, so in order for P’(x) – 2xP(x) to have order 5 as well P(x) needs to have order 4.

Note that x⁵ - (a² + b²)x³ + a²b²x has no x⁴ or x² terms so our P(x) should have no x³ or x terms to avoid x⁴ or x² terms cropping up in P’(x) – 2xP(x).

Thus P(x) = αx⁴ + βx² + γ for some α, β, γ to be determined.

P’(x) – 2xP(x) = (4αx³ + 2βx) – (2αx⁵ + 2βx³ + 2γx) = –2αx⁵ + (4α - 2β)x³ + (2β - 2γ)x

And now we equate coefficients:

–2αx⁵ + (4α - 2β)x³ + (2β - 2γ)x ≡ x⁵ - (a² + b²)x³ + a²b²x

       •   –2α = 1 ⇒ α = -0.5

       •   4α - 2β = - a² - b²  ⇒ -2 - 2β = - a² - b² ⇒ β = (a² + b² - 2)/2

       •   2β - 2γ = a²b² ⇒ a² + b² - 2 - 2γ = a²b² ⇒ γ = (a² + b²-  a²b² - 2)/2

Hence P(x) = -0.5x⁴ + (a² + b² - 2)x²/2 + (a² + b²-  a²b² - 2)/2 is a solution. 

George B. GCSE Maths tutor, A Level Maths tutor, A Level Further Math...

1 year ago

Answered by George, an Uni Admissions Test .STEP. tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist

12 SUBJECT SPECIALISTS

£25 /hr

Guy P.

Degree: Mathematics (Masters) - Warwick University

Subjects offered: .STEP., Maths+ 2 more

.STEP.
Maths
Further Mathematics
.MAT.

“About:Hi. I am a 2nd Year Mathematics student at the University of Warwick. I achieved a comfortable First in Year 1 and have continued this trend into my second year. Even from an early age, I have had a burning passion to engage m...”

£28 /hr

Vandan P.

Degree: Physical Natural Sciences (Bachelors) - Cambridge University

Subjects offered: .STEP., Science+ 6 more

.STEP.
Science
Physics
Maths
Further Mathematics
Chemistry
.MAT.
-Oxbridge Preparation-

“I am a second year undergraduate at Corpus Christi College Cambridge. I am studying Natural Sciences. In my first year I did Physics, Chemistry, Computer Science and Maths and passed with first class honours....”

£28 /hr

George B.

Degree: Mathematics (Masters) - Warwick University

Subjects offered: .STEP., Maths+ 1 more

.STEP.
Maths
Further Mathematics

“Third year undergraduate at one of the top universities for Maths. Eager to tutor and help improve your grades.”

About the author

£28 /hr

George B.

Degree: Mathematics (Masters) - Warwick University

Subjects offered: .STEP., Maths+ 1 more

.STEP.
Maths
Further Mathematics

“Third year undergraduate at one of the top universities for Maths. Eager to tutor and help improve your grades.”

You may also like...

Posts by George

Given that a and b are distinct positive numbers, find a polynomial P(x) such that the derivative of f(x) = P(x)e^(−x²) is zero for x = 0, x = ±a and x = ±b, but for no other values of x.

How do I solve Hannah’s sweet question?

Let P(z) = z⁴ + az³ + bz² + cz + d be a quartic polynomial with real coefficients. Let two of the roots of P(z) = 0 be 2 – i and -1 + 2i. Find a, b, c and d.

What values of θ between 0 and 2π satisfy the equation cosec(θ) + 5cot(θ) = 3sin(θ)?

Other Uni Admissions Test .STEP. questions

Given that a and b are distinct positive numbers, find a polynomial P(x) such that the derivative of f(x) = P(x)e^(−x²) is zero for x = 0, x = ±a and x = ±b, but for no other values of x.

Show that substituting y = xv, where v is a function of x, in the differential equation "xy(dy/dx) + y^2 − 2x^2 = 0" (with x is not equal to 0) leads to the differential equation "xv(dv/dx) + 2v^2 − 2 = 0"

By use of calculus, show that x − ln(1 + x) is positive for all positive x.

Prove: If pq, or p + q is irrational, then at least one of p and q is irrational.

View Uni Admissions Test .STEP. tutors

Cookies:

We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok