For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x)

Q: Wait, why do we assume $f'(1)=0$? Can't the derivative be non-zero?

Remember the factor theorem! If $(x-a)^k$ is a factor of $f(x)$, then $(x-a)^{k-1}$ is a factor of $f'(x)$. Since $(x-1)^2$ is a factor, $f'(1)$ must be zero. Geometrically, this means the curve is tangent to the x-axis at $x=1$ and $x=-1$.

Q: Why are we looking for the *minimum* value? Does $h(x)$ change things?

Great question! The problem asks for the minimum value. We assume the simplest case for $h(x)$ (like a constant) to ensure we don't accidentally create *more* roots than necessary. We are seeking the floor of the root count, so we look for the most constrained scenario.

Q: Is it possible for $f''(x)$ to have more than 2 roots?

Absolutely! But we are constrained by the minimum. By using Rolle's Theorem, we guarantee the existence of at least two roots. Since we want the minimum, we focus on these guaranteed roots.

Question

For a polynomial $g(x)$ with real coefficient, let $m_g$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficient defined by $S = \{(x^2 - 1)^2 (a_0 + a_1 x + a_2 x^2 + a_3 x^3) : a_0, a_1, a_2, a_3 \in \mathbb{R}\}$. For a polynomial $f$, let $f'$ and $f''$ denote its first and second order derivatives, respectively. Then the minimum possible value of $(m_{f'} + m_{f''})$, where $f \in S$, is _______

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given a set $S$ of polynomials defined by $f(x) = (x^2-1)^2 h(x)$, where $h(x)$ is a polynomial of degree at most $3$. Our objective is to determine the minimum possible value of the sum of the number of distinct real roots of the first and second derivatives, denoted as $m_{f'} + m_{f''}$.

The factor $(x^2-1)^2$ can be expanded as $(x-1)^2 (x+1)^2$. This structure implies that the curve is tangent to the $x$-axis at $x=1$ and $x=-1$.

In the language of calculus, this implies:
$$f(1) = 0, \quad f(-1) = 0, \quad f'(1) = 0, \quad 	ext{and} \quad f'(-1) = 0$$
These repeated roots force the first derivative to vanish at these specific points.

### The First Derivative and Rolle's Theorem

We have already established that $f'(x)$ has roots at $x = -1$ and $x = 1$. According to **Rolle's Theorem**, since $f(-1) = 0$ and $f(1) = 0$, there must exist at least one point $\alpha \in (-1, 1)$ such that $f'(\alpha) = 0$.

Consequently, we have identified at least three distinct real roots for $f'(x)$: $-1$, $\alpha$, and $1$. This leads us to the inequality:
$$m_{f'} \ge 3$$

### The Second Derivative and Inflection Points

Now, we apply **Rolle's Theorem** to the function $f'(x)$ to analyze $f''(x)$. We know that $f'(x)$ has roots at $-1$, $\alpha$, and $1$.

On the interval $[-1, \alpha]$, since $f'(-1) = 0$ and $f'(\alpha) = 0$, there must exist a point $\beta_1 \in (-1, \alpha)$ such that $f''(\beta_1) = 0$. Similarly, on the interval $[\alpha, 1]$, since $f'(\alpha) = 0$ and $f'(1) = 0$, there must exist a point $\beta_2 \in (\alpha, 1)$ such that $f''(\beta_2) = 0$.

We have thus identified at least two distinct real roots for $f''(x)$, which gives us:
$$m_{f''} \ge 2$$

### Final Calculation

We have determined that the minimum number of distinct real roots for the first derivative is $3$, and for the second derivative is $2$. Summing these values, we obtain:
$$m_{f'} + m_{f''} \ge 3 + 2 = 5$$

The beauty of this result lies in its independence from the specific coefficients of $h(x)$. The geometric constraints imposed by the factor $(x^2-1)^2$ dictate that the minimum value of the sum is **5**.

Enter Numerical Value:

Visualized Solution

Understanding the Polynomial Set $S$

Identifying the Roots of $f (x)$

First Derivative at Repeated Roots

Applying Rolle's Theorem for $f (x)$

Minimum Roots of $f^{'} (x)$

Rolle's Theorem for $f^{''} (x)$ on $[- 1, α]$

Rolle's Theorem for $f^{''} (x)$ on $[α, 1]$

Minimum Roots of $f^{''} (x)$

Final Calculation

The Sigma Insight: Mean Value Theorems

Analyzing the Setup

The First Derivative and Rolle's Theorem

The Second Derivative and Inflection Points

Final Calculation

Similar Questions

Let $f : R \to R$ be a thrice differentiable function such that $f (0) = 0, f (1) = 1, f (2) = - 1, f (3) = 2$ and $f (4) = - 2$ . Then, the minimum number of zeros of $(3 f^{'} f^{''} + f f^{'''}) (x)$ is _______

For a twice differentiable function $f (x), g (x)$ is defined as $g (x) = (f^{'} (x))^{2} + f^{''} (x) f (x)$ on $[a, e]$ . If for $a < b < c < d < e, f (a) = 0, f (b) = 2, f (c) = - 1, f (d) = 2, f (e) = 0$ then find the minimum number of zeros of $g (x)$ .

Let $f : (a, b) \to R$ be twice differentiable function such that $f (x) = \int_{a}^{x} g (t) d t$ for a differentiable function $g (x)$ . If $f (x) = 0$ has exactly five distinct roots in $(a, b)$ , then $g (x) g^{'} (x) = 0$ has at least :

Let f and g be twice differentiable even functions on (-2, 2) such that $f (\frac{1}{4}) = 0, f (\frac{1}{2}) = 0, f (1) = 1$ and $g (\frac{3}{4}) = 0, g (1) = 2$ Then, the minimum number of solutions of $f (x) g^{''} (x) + f^{'} (x) g^{'} (x) = 0$ in (-2, 2) is equal to ______.

Let $g : R \to R$ be a non constant twice differentiable such that $g^{'} (\frac{1}{2}) = g^{'} (\frac{3}{2})$ . If a real valued function $f$ is defined as $f (x) = \frac{1}{2} [g (x) + g (2 - x)]$ , then

Let $f : [0, 1] \to R$ be a twice differentiable function in $(0, 1)$ such that $f (0) = 3$ and $f (1) = 5$ . If the line $y = 2 x + 3$ intersects the graph of $f$ at only two distinct points in $(0, 1)$ , then the least number of points $x \in (0, 1)$ , at which $f^{''} (x) = 0$ , is ________

Let $f$ be a twice differentiable function on $(1, 6)$ . If $f (2) = 8$ , $f^{'} (2) = 5$ , $f^{'} (x) \geq 1$ and $f^{''} (x) \geq 4$ for all $x \in (1, 6)$ , then :

Let $f$ be any function continuous on $[a, b]$ and twice differentiable on $(a, b)$ . If for all $x \in (a, b)$ , $f^{'} (x) > 0$ and $f^{''} (x) < 0$ , then for any $c \in (a, b)$ , $\frac{f ( c ) - f ( a )}{f ( b ) - f ( c )}$ is greater than:

For all $x$ in $[0, 1]$ , let the second derivative $f^{''} (x)$ of a function $f (x)$ exist and satisfy $∣ f^{''} (x) ∣ < 1$ . If $f (0) = f (1)$ , then show that $∣ f^{'} (x) ∣ < 1$ for all $x$ in $[0, 1]$ .

$f (x)$ is a differentiable function and $g (x)$ is a double differentiable function such that $∣ f (x) ∣ \leq 1$ and $f^{'} (x) = g (x)$ . If $f^{2} (0) + g^{2} (0) = 9$ . Prove that there exists some $c \in (- 3, 3)$ such that $g (c) g^{''} (c) < 0$ .

Enter Numerical Value:

Visualized Solution

Understanding the Polynomial Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S

Identifying the Roots of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)

First Derivative at Repeated Roots

Applying Rolle's Theorem for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)

Minimum Roots of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f′(x)

Rolle's Theorem for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f′′(x) on .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } [−1,α]

Rolle's Theorem for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f′′(x) on .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } [α,1]

Minimum Roots of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f′′(x)

Final Calculation

The Sigma Insight: Mean Value Theorems

Analyzing the Setup

The First Derivative and Rolle's Theorem

The Second Derivative and Inflection Points

Final Calculation

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let g:R→R be a non constant twice differentiable such that g′(21​)=g′(23​). If a real valued function f is defined as f(x)=21​[g(x)+g(2−x)], then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f be a twice differentiable function on (1,6). If f(2)=8, f′(2)=5, f′(x)≥1 and f′′(x)≥4 for all x∈(1,6), then :

Understanding the Polynomial Set $S$

Identifying the Roots of $f (x)$

Applying Rolle's Theorem for $f (x)$

Minimum Roots of $f^{'} (x)$

Rolle's Theorem for $f^{''} (x)$ on $[- 1, α]$

Rolle's Theorem for $f^{''} (x)$ on $[α, 1]$

Minimum Roots of $f^{''} (x)$

Let $g : R \to R$ be a non constant twice differentiable such that $g^{'} (\frac{1}{2}) = g^{'} (\frac{3}{2})$ . If a real valued function $f$ is defined as $f (x) = \frac{1}{2} [g (x) + g (2 - x)]$ , then

Let $f$ be a twice differentiable function on $(1, 6)$ . If $f (2) = 8$ , $f^{'} (2) = 5$ , $f^{'} (x) \geq 1$ and $f^{''} (x) \geq 4$ for all $x \in (1, 6)$ , then :